Datasets:

jbrinkma

/

pile-100k

like 0

Introduction {#Sec1} ============ Structured light, one kind of special optical beams with spatial inhomogeneous intensity, polarization or phase distributions, turns over a new leaf in the development of modern optics^[@CR1]--[@CR6]^. In addition to the conventional frequency and amplitude modulation, the creation of structured light including optical vortex and vector beam can as well exploit the complex phase and polarization of light, pushing the capacity of light manipulation closer to full potentials. This capability of tailoring additional dimensions on light finds a perfect synergy with the optical nanotechnology, breeding a plethora of splendid possibilities in integrated photonic circuits, quantum communication, nano-holography, and near-field sensing^[@CR7]--[@CR13]^. However, it is typically difficult to simultaneously and effectively manipulate polarization and phase of light at nanoscale which is critical in ultra-compact integrated optics, because the optical anisotropy provided by naturally existing materials is generally too finite to significantly modulate light in deep-subwavelength space. The emergence of metamaterials and metasurfaces contributes a solution to this difficulty^[@CR3]^. Metamaterials and metasurfaces are artificial subwavelength-structured composite materials with engineered electromagnetic properties on demand that can break nature's limit, including artificial magnetism, tailored anisotropy, negative index of refraction, near zero permittivity or index, and thus can provide sufficient control over light as required^[@CR14]--[@CR22]^. A typical approach for simultaneous manipulation of light polarization and phase is by transmitting incident beam through an anisotropic medium, where the two originally independent physical quantities are coupled and connected. The relationship between the polarization and phase variation of the transmitted beam can be described by the space-domain geometric phase---'the Pancharatnam-Berry phase'^[@CR23],[@CR24]^, discovered by the pioneer scientists^[@CR25]--[@CR29]^. Based on this principle, the anisotropic optical element, termed as the Pancharatnam-Berry phase Optical Element (PBOE), can be used to induce the designated phase and polarization manipulations. To date, a plenty of complex light manipulation including the generation of scalar and vector vortices have been enabled by the smartly designed metamaterial or metasurface based PBOEs in special orientations and patterns including nano-rods, nano-gratings, V-shape antennas, split-ring antennas, and multilayer cavities^[@CR9],[@CR30]--[@CR40]^. Among many kinds of PBOEs, here we are particularly interested in the optical element that can induce quarter-wave retardation between the two orthogonal linear polarization axes, acting as a quarter-wave plate (QWP). Such element has the unique property that can convert circular polarization (CP) into linear polarization (LP) with the polarization direction depending on the QWP orientation. Structured polarization and phase then can be created by the spatially inhomogeneous array of the QWP elements under the PBOE principle. In this work, we propose and demonstrate the magnetic metamaterial quarter-wave turbines at visible wavelength to twist circularly polarized incident beam into radially and azimuthally polarized vector vortices. The magnetic metamaterial elements based on metal-dielectric-metal three-layer structures are designed to utilize both the electric and magnetic field of light and realize QWP functionality at the designated wavelength and maintain the quarter-wave retardation in broadband, inducing the broadband circular-to-linear polarization conversion. The turbine blades are made of multiple polar sections of the homogeneously oriented magnetic metamaterial QWP gratings with the fast axis of the QWP element in approximately +/−45° to the azimuthal angle. The constructed metamaterial quarter-wave turbines will transform the incident CP to vector polarization with helical phase front of charge 1. Furthermore, the vector polarization distributions of the produced vortices can be conveniently tuned azimuthally or radially by simply switching the spin of the incident CP beam or the turbine rotation direction. Results {#Sec2} ======= Design and characterization of magnetic metamaterial QWP gratings {#Sec3} ----------------------------------------------------------------- The homogeneous magnetic metamaterial gratings are designed, fabricated, and characterized to confirm its functionality as a good QWP, which will serve as the fundamental building blocks for constructing the spatial inhomogeneous magnetic metamaterial quarter-wave turbines. As shown in Fig. [1](#Fig1){ref-type="fig"}, a typical magnetic metamaterial grating structure contains a dielectric spacer sandwiched by two metal stripes in a subwavelength periodic pattern. The physical mechanism of such structure has been well studied by the previous literatures^[@CR38],[@CR41]--[@CR44]^. Magnetic resonances can be excited under transverse-magnetic (TM) illumination in this structure, where the magnetic field of the incident beam is parallel with the metal stripes. The incident magnetic field produces a circulating displacement current loop encircling the dielectric spacer with anti-parallel surface currents between the pair of metal stripes, which then excites a magnetic dipole moment normal to the current loop and parallel to the metal stripes. Magnetic resonance can then be excited causing a possibly negative effective permeability and an abrupt transmission phase variation across the resonance wavelength. Under transverse-electric (TE) polarized illumination where the electric field is parallel with the metal stripes, on the contrary, this structure resembles a diluted metal without supportable optical resonances, and has a generally flat transmission phase spectrum. As a result, the magnetic metamaterial gratings are able to manipulate both electric and magnetic fields of light at the same time, which can simultaneously control the transmission amplitude and phase in the orthogonal polarizations. Such control over the two orthogonal polarizations is important here. A perfect waveplate requires simultaneous phase anisotropy and amplitude isotropy. However, while the magnetic resonance induces the required phase retardation, it also reduces the amplitude transmission under TM polarization. This weakened amplitude transmission under TM polarization can then be balanced with the suppressed transmission by the grating effect under TE polarization to satisfy the amplitude isotropy requirement.Figure 1(**a**) The schematic of magnetic metamaterial gratings in a metal-dielectric-metal three-layer structure. (**b**) The simulated magnetic field (color map) and electric displacement (black arrows) on magnetic resonance under TM polarization. In this work, we design and fabricate the magnetic metamaterial grating samples to realize nearly ideal QWP devices operating near the wavelength of HeNe laser at 633 nm, with the excited magnetic resonance around 575 nm. The design principle is based on the physics of the magnetic metamaterials: the resonance wavelength is mainly determined by the width of the cavity and independent of the grating period; the phase retardation between TE and TM can be controlled by varying the thickness of each layer; the amplitude balance between TE and TM can be achieved by changing the grating period to modify the filling fraction of this structure. Eventually, by scanning these structural parameters of the structure in simulation, the optimized parameter set can be achieved for giving the best performance as QWP and such parameter set will be used in the sample fabrication. The final fabricated structure is as follows. The three-layer Ag-SiO~2~-Ag structure is deposited on glass slide by the sputtering method at the deposition rates of 0.4 Å/sec and 0.2 Å/sec for Ag and SiO~2~ layers, respectively, which has a thickness of 40 nm for each layer, together with an additional 5 nm thick SiO~2~ surface protection layer on top. The magnetic metamaterial grating pattern is then milled into the Ag-SiO~2~-Ag multilayer by focus ion beam (FIB) lithography with the optimized parameters of grating period as 300 nm and bottom metal stripe width as 140 nm. In all of the following experimental and modeling configurations, the samples are always placed so that the incident beam enters from the glass side to the multilayer side. The SEM image of the fabricated magnetic metamaterial grating sample is shown in Fig. [2(a)](#Fig2){ref-type="fig"}.Figure 2(**a**) The SEM images of the top-view and cross-section of the fabricated magnetic metamaterial grating sample. (**b**) The measured and simulated transmission spectra of the magnetic metamaterial grating sample. Numerical simulation of the proposed magnetic metamaterial QWP grating is performed with the finite element method COMSOL Multiphysics. The final geometries of the modeled structure, including a slightly tilted side-wall of the grating cross section, are obtained from the SEM image of the fabricated structure shown in Fig. [2(a)](#Fig2){ref-type="fig"}. The Ag and SiO~2~ permittivities are acquired from the variable angle spectroscopic ellipsometry. We adjusted the loss of the Ag in the simulation so that the simulated transmission spectrum fits the experimental transmission spectrum, according to the previous study on the similar structure^[@CR44]^ for the purpose of acquiring a more accurate estimation of the transmission phase. The simulated and measured transmission spectra of this structure are shown in Fig. [2(b)](#Fig2){ref-type="fig"}. It shows the distinctive feature of the modeled magnetic resonance under TM polarization is around 575 nm, where the circulating displacement current loop encircles around the magnetic dipole as shown in Fig. [1(b)](#Fig1){ref-type="fig"}. The magnetic resonance introduce a transmission minimum and an abrupt quarter-wave phase shift across the resonance under TM polarization, and the created phase difference is preserved stably above the resonance wavelength. This quarter-wave retardation and the proximity of equal TE and TM transmissions near 633 nm demonstrate the realization of a decent QWP with its fast-axis along TE direction. From Fig. [3(a)](#Fig3){ref-type="fig"}, it is interesting to observe that the phase retardation between TE and TM polarizations is nearly a constant around 90° after the magnetic resonance wavelength. Since there are no more resonances above 633 nm, this constant phase retardation is expected to be broadband extending to the longer wavelength side of 750 nm, until the relative thickness of this structure as compared to wavelength further decreases and can not support the required phase retardation. The nearly constant phase retardation should be attributed to the magnetic resonance supported in the longitudinal grating cavity, since such phenomenon does not exhibit in the structure of single metal layer. The semi-infinitely broadband constant quarter-wave phase retardation is a very desirable property, as this structure can serve as a broadband CP-to-LP converter. However, as the amplitude proximity is only satisfied in a limited bandwidth around 633 nm, the QWP function is specified in this particular narrow wavelength region.Figure 3(**a**) The simulated transmission phase spectra of the magnetic metamaterial grating. (**b**) The simulated spatial field phase distribution of the grating structure under TE and TM polarizations comparing with the air reference. Next, the performance of circular-to-linear polarization conversion by the magnetic metamaterial QWP grating is quantitatively characterized. It is emphasized that, as CP and LP are not mutually orthogonal, we cannot define energy-based polarization conversion efficiency between them. Instead, we illustrate such polarization conversion by inspecting the degree of linear polarization (DoLP) and the angle of linear polarization (AoLP) of the transmitted beam under CP incidence in both experiment and simulation. The DoLP is defined by the polarization Stokes parameters expressed in Equation ([1](#Equ1){ref-type=""}), and the AoLP represents for the orientation angle of the dominant linear polarization component^[@CR17],[@CR20],[@CR45]^.$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$DoLP=\frac{\sqrt{{s}_{1}^{2}+{s}_{2}^{2}}}{{s}_{0}}$$\end{document}$$Where *s* ~0~ *s* ~1~ and *s* ~2~ are the corresponding Stokes parameters. Here, by considering the magnetic metamaterial gratings as effective media, the transmitted beam from CP incidence will have a general elliptical polarization. Its Stokes parameters, in experiment, can be obtained based on the polarization ellipse, which is characterized by the classical polarization analysis experiment^[@CR17],[@CR46]^ as shown in Fig. [4](#Fig4){ref-type="fig"}. In this experiment, a CP beam is first created by passing the white light beam from a halogen lamp through a linear polarizer and a quarter-wave plate which has a 45° tilted angle between their axes, then the beam is normally focused through the magnetic metamaterial grating sample with the stripes oriented vertically. Next, the transmitted beam from the sample passes through a rotating linear polarization analyzer, and the spectra of the final beam with angular-response are measured by a spectrometer. The transmission spectra are normalized by the intensity of the beam before passing the analyzer. The corresponding polarization ellipse parameters at each wavelength can be retrieved by a sinusoidal fitting function shown in Equation ([2](#Equ2){ref-type=""}).$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(\psi )={a}^{2}co{s}^{2}(\psi -\theta )+{b}^{2}si{n}^{2}(\psi -\theta )$$\end{document}$$where *ψ* represent for the angle of polarization analyzer, *a* and *b* represent for the long and short radius of the polarization ellipse, and *θ* represent for the tilted angle of the ellipse (AoLP). Then the Stokes parameters can be calculated by the following equations related to the electric field components *E* ~*a*~ and *E* ~*b*~ ^[@CR47]^.$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}{s}_{0}={E}_{a}^{2}+{E}_{b}^{2}\\ {s}_{1}=({E}_{a}^{2}-{E}_{b}^{2})cos(2\theta )\\ {s}_{2}=({E}_{a}^{2}-{E}_{b}^{2})sin(2\theta )\end{array}$$\end{document}$$ Figure 4(**a**) The schematic of the experimental setup for polarization analysis. (**b**) The measured polarization analysis spectra with different rotation angle of the linear polarization analyzer. LP: linear polarizer, MMs: metamaterials. Based on the analysis from Equations ([1](#Equ1){ref-type=""})--([3](#Equ3){ref-type=""}), the DoLP and AoLP in experiment are derived and plotted in Fig. [5(a)](#Fig5){ref-type="fig"}.Figure 5(**a**,**b**) The DoLP and AoLP retrieved from experiment and simulation, respectively. In simulation, the magnetic metamaterial grating unitcell under the same CP illumination is modeled, from which the normalized field transmittance can be acquired. Then the Stokes parameters and the tilted angle of the polarization ellipse are calculated from Equation ([4](#Equ4){ref-type=""}) below^[@CR47]^.$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}{s}_{0}={|{E}_{x}|}^{2}+{|{E}_{y}|}^{2}\\ {s}_{1}={|{E}_{x}|}^{2}-{|{E}_{y}|}^{2}\\ {s}_{2}=2Re({E}_{x}{E}_{y}^{\ast })\\ \theta =\frac{1}{2}{\rm{atan}}(\frac{2{E}_{x}{E}_{y}}{{E}_{x}^{2}-{E}_{y}^{2}}\,\cos (\delta ))\end{array}$$\end{document}$$ Figure [5(b)](#Fig5){ref-type="fig"} plots the simulated DoLP and AoLP, which exhibits a decent agreement with the experimental results. As shown in Fig. [5](#Fig5){ref-type="fig"}, the DoLP and AoLP spectra demonstrate the magnetic metamaterial grating sample operates as an excellent QWP near 633 nm, where the transmitted beam has the DoLP and AoLP close to unity and 45°, respectively. It is remarkable that the function of QWP can be realized by many different metamaterial and metasurface structures as illustrated by the previous research^[@CR17],[@CR19]--[@CR21]^. Besides, other works study optical HWP element using dielectric nanostructures with near-unity polarization (spin) conversion efficiency and minimum optical loss^[@CR48]--[@CR50]^. Analogous ideas could also be applied to make dielectric QWP element with similarly excellent performance. Comparing to these works, the most significant feature of our current structure is such magnetic metamaterial grating maintains the quarter-wave retardation in a broadband wavelength range in direct transmission mode, thus it can produce a highly linearly polarized beam with DoLP larger than 95%, above the magnetic resonance wavelength in broadband. Further, our structure has a relatively thin thickness as one quarter of wavelength, and it is convenient to fabricate as a one-dimensional grating structure, in contrast to the reported dielectric-post waveplate which has a thickness close to the wavelength and the thickness is larger than the period. These merits can be an important advantage at least for now, as fabricating a structure with larger longitudinal-to-transverse aspect ratio is generally more difficult especially for devices operating in visible frequency range. Vector vortex generation by magnetic metamaterial quarter-wave turbines {#Sec4} ----------------------------------------------------------------------- By combining the magnetic metamaterial QWP gratings with certain orientations as the building blocks, the spatial inhomogeneous quarter-wave turbines are constructed with each grating section as one turbine blade, as shown in the SEM images of the fabricated samples in Fig. [6](#Fig6){ref-type="fig"}. These metamaterial quarter-wave turbines are designed to create vector vortices from CP incident beam. The whole turbine pattern is segmented into multiple polar sections, which individually has the uniformly oriented magnetic metamaterial QWP grating with its fast-axis in approximate +/−45° to the radial-direction for constructing the counter-clockwise rotation turbine or the clockwise rotation turbine, respectively. Therefore, the QWP gratings within each section will convert the incident CP beam into LP beam with its polarization direction in +/−45° to the QWP fast-axis, making an overall radial or azimuthal polarization for the final transmitted beam through the entire turbine. According to the PBOE principle, a rotation distribution of CP-to-LP polarization conversion element in a full 2*π* around an origin will produce a charge +1 or −1 optical vortex from CP incidence^[@CR27],[@CR29],[@CR38]^. Here, the generation of optical vortices by the turbines is broadband due to the broadband CP-to-LP conversion of the magnetic metamaterial grating, although outside the optimized QWP wavelength region near 633 nm the polarization state of the transmitted beam will not be purely radial or azimuthal but their hybrid.Figure 6(**a**,**b**) The SEM image of the fabricated magnetic metamaterial quarter-wave turbines with counter-clockwise rotation and clockwise rotation, respectively. (**c**) The simulated counter-clockwise rotation turbine structure. The mathematical derivation of vector vortex generation by the ideal quarter-wave turbines is as follows. First, the CP incident beam can be expressed in the cylindrical vector coordinates. Taking the left-handed circular polarized (LHC) beam for example, the electric field is$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}{E}_{LHC} & = & P(r)\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{x}}}+i{{\boldsymbol{e}}}_{{\boldsymbol{y}}})\\ & = & P(r)\frac{\sqrt{2}}{2}((\cos (\phi ){{\boldsymbol{e}}}_{{\boldsymbol{r}}}-\,\sin (\phi ){{\boldsymbol{e}}}_{\phi })\\ & & +i(\sin (\phi ){{\boldsymbol{e}}}_{{\boldsymbol{r}}}+\,\cos (\phi ){{\boldsymbol{e}}}_{\phi }))\\ & = & P(r)\frac{\sqrt{2}}{2}{e}^{i\phi }({{\boldsymbol{e}}}_{{\boldsymbol{r}}}+i{{\boldsymbol{e}}}_{\phi })\end{array}$$\end{document}$$where $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(r)$$\end{document}$ is the spatial amplitude profile, *φ* is the azimuthal angle, ***e*** ~***r***~ and ***e*** ~***φ***~ are unit vectors in the radial and azimuthal direction, respectively. Then, define a new coordinate system with two axes that have a +/−45° rotation to the radial direction, in which the pair unit vectors are defined as $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{e}}}_{{\boldsymbol{r}}+{45}^{\circ }}$$\end{document}$ and $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{e}}}_{{\boldsymbol{r}}-{45}^{\circ }}$$\end{document}$ respectively, as $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{e}}}_{{\boldsymbol{r}}+{45}^{\circ }}=\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}}-{{\boldsymbol{e}}}_{\phi })$$\end{document}$, and $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{e}}}_{{\boldsymbol{r}}-{45}^{\circ }}=\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}}+{{\boldsymbol{e}}}_{\phi })$$\end{document}$, so that$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}{{\boldsymbol{e}}}_{{\boldsymbol{r}}}=\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}+{45}^{\circ }}+{{\boldsymbol{e}}}_{r-{45}^{\circ }}),\\ {{\boldsymbol{e}}}_{\phi }=\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}+{45}^{\circ }}-{{\boldsymbol{e}}}_{r-{45}^{\circ }})\end{array}$$\end{document}$$ Then the LHC beam can be expressed in the corresponding *r* + 45° and *r* − 45° coordinates as:$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}{E}_{LHC} & = & P(r)\frac{\sqrt{2}}{2}{e}^{i\phi }(\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}{\boldsymbol{+}}4{{\bf{5}}}^{\circ }}+{{\boldsymbol{e}}}_{{\boldsymbol{r}}-{45}^{\circ }})+i\frac{\sqrt{2}}{2}({{\boldsymbol{e}}}_{{\boldsymbol{r}}{\boldsymbol{+}}4{{\bf{5}}}^{\circ }}-{{\boldsymbol{e}}}_{r-{45}^{\circ }}))\\ & = & \frac{1}{2}P(r){e}^{i\phi }((1+i){{\boldsymbol{e}}}_{{\boldsymbol{r}}{\boldsymbol{+}}4{{\bf{5}}}^{\circ }}+(1-i){{\boldsymbol{e}}}_{{\boldsymbol{r}}{\boldsymbol{-}}4{{\bf{5}}}^{\circ }})\end{array}$$\end{document}$$ After the LHC beam passing through the clockwise rotation turbine, which is treated as the QWP with its fast axis at −45° to the radial direction, the transmitted beam can be expressed as:$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}{E}_{out} & = & \frac{1}{2}\,P(r){e}^{i\phi }t\,((1+i){{\boldsymbol{e}}}_{{\boldsymbol{r}}+{\bf{4}}{{\bf{5}}}^{\circ }}+(1-i){{\boldsymbol{e}}}_{r-{\bf{4}}{{\bf{5}}}^{\circ }})(\begin{array}{c}1\\ i\end{array})\\ & = & \frac{\sqrt{2}}{2}(1+i)t{e}^{i\phi }{{\boldsymbol{e}}}_{r}\end{array}$$\end{document}$$where *t* is the amplitude transmittance of the QWP element. This expression shows that the transmitted beam has the radial polarization with a charge +1 orbital angular momentum (OAM). The circumstances with an arbitrary combination of incident spin and turbine rotation direction can be calculated accordingly with the following conclusions for the phase and polarization manipulations. First, LHC incident beam will always generate charge +1 OAM, while RHC incident beam will always generate charge −1 OAM, where spin-to-orbital angular momentum conservation is satisfied. Second, radial polarization will be produced for LHC input with clockwise rotation turbine or RHC input with counter-clockwise rotation turbine; while azimuthal polarization will be produced for LHC input with counter-clockwise rotation turbine and RHC input with clockwise rotation turbine. Concerning the general method of vector vortex generation by polarization conversion under the PBOE principle, it is noteworthy that the linear polarizer element in special patterns such as concentric rings can also achieve similar functionality as illustrated by some previous works^[@CR31],[@CR38],[@CR51]^. However, the QWP element, as a comparison, has two distinguished merits. First, the QWP element exploits the complete incident field while the linear polarizer element instantaneously blocks half of the field, which grants the QWP element to provide higher energy efficiency. Second, the vector polarization produced by the QWP element can be conveniently switched from radial to azimuthal by inversing the incident spin or switching the QWP fast-axis orientation. Next, the 3D simulation of the magnetic metamaterial quarter-wave turbine is performed by using the COMSOL Multiphysics. The simulated structure is shown in Fig. [6(c)](#Fig6){ref-type="fig"}, where only the inner-most eight polar sections of the counter-clockwise rotation turbine are included. The structural and material parameters of each grating unitcell in the 3D simulation are acquired from those used in the previous simulation for the magnetic metamaterial grating structure, and the illumination field has been set as either LHC or RHC beam. The phase profile and polarization distribution of the transmitted beam at near-field are calculated and plotted in Fig. [7](#Fig7){ref-type="fig"}, which clearly indicate the vector vortex generation as expected.Figure 7(**a**,**c**) The simulated phase distribution of the generated vector vortices at near-field with LHC and RHC incident beam transmitted through the counter-clockwise rotation turbine, creating charge +1 and −1 OAM, respectively. (**b**,**d**) The simulated intensity profile (color map) and polarization distribution (black arrows) of the generated vector vortices with LHC and RHC incident beam through the counter-clockwise rotation turbine, producing azimuthal or radial polarization profiles, respectively. Based on the fabrication parameters of the homogeneous magnetic metamaterial gratings, the metamaterial quarter-wave turbine samples with both counter-clockwise rotation and clockwise rotation are fabricated and characterized. The total fabrication area for each turbine sample is a disk with a 24 *μ*m-diameter, which is divided into four quadrants during the FIB milling. There are slightly stitching errors at the boundary between adjacent quadrants. The SEM images of the fabricated counter-clockwise rotation turbine and clockwise rotation turbine are shown in Fig. [6](#Fig6){ref-type="fig"}. The characterization experimental setup is shown in Fig. [8](#Fig8){ref-type="fig"}, where a CP incident beam output from a 633 nm HeNe laser through a linear polarizer and a QWP is normally focused on the turbine sample. Then the transmitted beam is re-collimated and characterized by spherical wave interferometry and polarization analysis for studying the output OAM and polarization distribution.Figure 8The schematic of characterization experimental setup for the transmission imaging, spherical wave interferometry and polarization analysis of the generated vector vortices through the magnetic metamaterial quarter-wave turbine sample. OL: objective lens, BS: beam splitter, LP: linear polarizer. The experimental results are summarized in Figs [9](#Fig9){ref-type="fig"}--[12](#Fig12){ref-type="fig"}, for all the combinations of incident spin of LHC/RHC and turbine counter-clockwise/clockwise rotation direction. The interference patterns in all cases exhibit single spiral fringes, which has counter-clockwise orientation for LHC incidence or clockwise orientation for RHC incidence, indicating either charge +1 or −1 OAM is obtained in the output vector vortex beam. The transmission images after the linear polarization analyzer always exhibit a two-lobe shape, which rotates together with the rotation angle of the polarization analyzer. The azimuthal or radial polarizations are signified when the darkline between the two lobes is in parallel or perpendicular with the polarization analyzer direction, respectively. The polarization analysis results present that azimuthally polarized vector vortices are generated when LHC (or RHC) incident beam transmits through counter-clockwise (or clockwise) rotation turbine, while radially polarized vector vortices are produced when LHC (or RHC) incident beam transmits through clockwise (or counter-clockwise) rotation turbine. These experimental observations further confirm the theoretic and modeling results explained previously.Figure 9The generated azimuthally polarized vector vortex with charge +1 from the counter-clockwise rotation turbine under LHC input. (**a**),(**b**) The direct transmission images and the interference pattern of the vector vortex, respectively. (**c**) The transmission images after the vector vortex passing a rotating linear polarization analyzer, where the white arrows represent for the orientation angles of the polarization analyzer from 0 to 7π/8. Figure 10The generated radially polarized vector vortex with charge −1 from the counter-clockwise rotation turbine under RHC input. (**a**),(**b**) The direct transmission images and the interference pattern of the vector vortex, respectively. (**c**) The transmission images after the vector vortex passing a rotating linear polarization analyzer. Figure 11The generated radially polarized vector vortex with charge +1 from the clockwise rotation turbine under LHC input. (**a**),(**b**) The direct transmission images and the interference pattern of the vector vortex, respectively. (**c**) The transmission images after the vector vortex passing a rotating linear polarization analyzer. Figure 12The generated azimuthally polarized vector vortex with charge −1 from the clockwise rotation turbine under RHC input. (**a**,**b**) The direct transmission images and the interference pattern of the vector vortex, respectively. (**c**) The transmission images after the vector vortex passing a rotating linear polarization analyzer. Discussions {#Sec5} =========== Lastly, we discuss some important principles and outcomes about our proposed concept and demonstrated results. It is noteworthy that the cross-coupling between neighboring PBOEs has a substantial impact on the overall PBOE array design. For weakly coupled optical elements in which the transmission is independent on their periodicity orientation, such like low filling-fraction plasmonic or dielectric nano-antennas, desired phase manipulation can be induced by simply rotating their functional part (antenna) without rotating periodicity^[@CR34],[@CR37],[@CR40]^. Spatial inhomogeneous PBOE array based on such weakly coupled PBOEs is most convenient to the designs with excellent flexibility and accuracy. The magnetic metamaterials studied here, on the other hand, are the strongly coupled grating structures, and we must rotate the whole unitcell in order to effectively control the transmitted phase from the PBOE principle. Thus, except for some very special patterns such as concentric rings, it is almost impossible to make spatial inhomogeneous patterns while perfectly maintaining the periodicity for strongly coupled structures. This is a setback. However, on the other hand, strongly coupled structures usually have stronger interaction over the incident beam, thus can realize designated function as waveplate or polarizer with thinner device thickness. For example, the magnetic metamaterial QWPs here realize the QWP function with the total thickness less than one quarter of the wavelength, comparing to the weakly coupled dielectric slab structure which has the thickness close to one wavelength^[@CR48]--[@CR50]^. A thinner optical thickness makes such structure a closer approximation as "optical surface" with more robust tolerance to oblique incidence, very desirable in many optical applications. Furthermore, as mentioned before, our magnetic metamaterial grating structure has a nearly 50% filling fraction and an approximate 1:2.5 height-to-period aspect ratio which makes it convenient to fabricate. We developed the polar sectioned grating array to address the periodicity rotating difficulty. Each polar section contains uniform grating unitcells whose periodicity is perfectly preserved. But the boundaries between adjacent sections bring slight error to light manipulation. Naturally the denser the sections, the more accurate the phase and polarization manipulation can be achieved. However, each section should contain enough number of grating periods to make it physically effective for the incident beam. Eventually we design each polar section to contain about 10 grating periods. Another noteworthy point is that the intensity profiles of the generated vortices exhibit a certain degree of asymmetry. There may be some different contributing factors, among which the most significant reason, in our understanding, is due to the stitching error in the fabrication process. The geometrical errors in the section boundaries from design and the stitching error from fabrication together induce diffractions of the transmitted beam, causing a split from the eigenmode of the optical vortex and eventually produce non-integer OAM and hybrid polarization states in the output. Previous studies suggest that mixed modes of optical vortices are not stable^[@CR1]^, and are susceptible to split into composite singularities in far field, which can induce an asymmetry intensity profile^[@CR52],[@CR53]^. The stitching error between the adjacent quadrants shown in Fig. [6(a,b)](#Fig6){ref-type="fig"} is due to the alignment shift in the FIB tool when we fabricate the quadrants sequentially. Naturally, such error could be eliminated if the complete fabrication pattern can be loaded into the FIB tool as one bitmap. Previous work shows highly symmetric vortex can be generated via a similar structure from a perfect fabrication by a powerful tool^[@CR54]^. This intensity asymmetry in the current work, on the other hand, can be cleaned by applying suitable optical filter such like few-mode fiber in post-processes^[@CR55]^. Finally, we emphasis that the magnetic metamaterial QWPs in the +/−45° turbine blades presented here is just one example of using multiple polar sections to enable spatial inhomogeneous light manipulation under PBOE principle. This particular structure can only produce +1/−1-charge radially or azimuthally polarized beam; however, through a different unitcell and orientation inside the polar sections, arbitrary light phase and polarization control can be achieved based on the optically-thin strongly coupled nanostructures. Conclusions {#Sec6} =========== In summary, we design, fabricate and characterize spatial inhomogeneous magnetic metamaterial quarter-wave turbines to create radially and azimuthally polarized vector vortices from circularly polarized incident beam at visible wavelength. The metamaterial turbines are constructed from the sectioned magnetic metamaterial gratings oriented along the specified angles exhibiting excellent QWP feature in the assigned wavelength and broadband efficient CP-to-LP conversion response. Following the PBOE principle, the metamaterial quarter-wave turbines are able to convert the CP incident beam into vector vortex beam with spin-dependent charge +1 or −1 OAM. Moreover, the switch between the azimuthally or radially polarized vector vortex beam is realized by simply reversing the incident spin or flipping the turbine rotation direction. Our demonstrated magnetic metamaterial quarter-wave turbine device is compact, efficient and tunable. It can be integrated into the future photonic and optoelectronic circuits for structured beam conversion, wavefront shaping, particle trapping and communications. Our proposed device will also serve as an inspirational example for exploring many other on-chip optical devices in complex polarization manipulation and phase shaping. **Publisher\'s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The authors acknowledge support from the Office of Naval Research under Grant No. N00014-16-1-2408, and the National Science Foundation under Grant No. DMR-1552871, ECCS-1653032, and CBET-1402743. The authors also acknowledge the facility support from the Materials Research Center at Missouri S&T. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by the Sandia Corporation, a wholly owned subsidiary of the Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. The author Jinwei Zeng is currently affiliated with the department of Electrical Engineering and Computer Science, University of California Irvine, Irvine, CA, 92697. J.Z. and X.Y. conceived the idea of this research. J.Z. performed all the numerical simulations and experiments. T.L. is responsible for the deposition of metal and dielectric layers. J.G. and X.Y. directed the research. All authors discussed the results and contributed to the manuscript. Competing Interests {#FPar1} =================== The authors declare that they have no competing interests.

Rosario Mendes Rosario Mendes (born 25 October 1989) is an Indian professional footballer who plays as a midfielder for Salgaocar in the I-League. Career Salgaocar Born in Cortalim, Goa, Mendes made his debut for Salgaocar F.C. in the I-League on 13 April 2013 against United Sikkim in which he came on in the 70th minute for Nicolau Colaco as Salgaocar won the match 9–0. Career statistics References External links Category:1989 births Category:Living people Category:Footballers from Goa Category:I-League players Category:Association football midfielders Category:Salgaocar F.C. players Category:People from South Goa district Category:Indian footballers

Menu How can homework help students As long as you are a student, you can never escape getting assigned homework by your tutors.Wedohomework.net provides assistance regarding different types of. At the Public Library Homework Help I had always looked for a homework organizer and reminder app but.Fellow students learn responsibility when they have homework. Child Struggling with Homework No matter what your homework help needs may be, 123Homework.com is here to.For that reason, assigning students some homework can be beneficial. However,.University homework help does not only help students achieve the best they can, but also strive on the feedback of our customers. Study Skills Tips. Online Homework Help Become computer Science Projects for Students The Best Student in Your Class. Sites to Help with Homework If you need urgent and professional help with science homework, our company can provide you with. Student Doing Homework Bottom line: students have too much homework and most of it is not productive or necessary. As you can imagine, that kind of homework rarely happens. Finance Homework Assignment Help Wud filigreed Abel superfuses pillwort how can homework help students deduced vanning dually. How Can Homework Help Students 72 Tips For Helping Kids and Teens With Homework and Study. can help to control the length. of time for homework. Students, teachers, parents, and everyone can find solutions to their math problems instantly.Parent help can backfire. expectations and help students develop a homework. Should Parents Help with Homework Online Homework Help Service For College and Graduate Students. The Homework Debate: How Homework Benefits Students. they complete homework that can help them. good habits in students, homework must prove useful. Black Parents Helping with Homework If they encounter stress in the form of parental agitation during homework, then where can.The results of such studies suggest that homework can improve students. Children and Parents Doing Homework Literacy Early Childhood Education Students and parents appear to carry similar critiques of homework,. App That Helps with Math Homework As a college student I get homework and yes it can be annoying but it reinforce the concepts you. Too Much Homework Stress Students The best multimedia instruction on the web to help you with your homework and study.

[Comparative experimental investigations with bioglass (L. L. Hench) and Al2O3-ceramic coated with mod. bioglass. II. Results of experiments with loaded implants (author's transl)]. This report is based on 4 comparative experimental series is sheep using a cementfree total hip replacement model. Basically a smooth stem design was compared to a step stem design for biomechanical interlocking anchorage. Both designs were implanted as pure high density Al2O3-ceramic components and as Al2O3-ceramic parts coated with Bioglass or Bioglass-ceramic (post op. follow up 13 months, 17 animals). None of the 4 stem modifications showed a tendency for permanent stabilisation or bonding in bone. Besides insufficient bonding of the glass coatings to the substrate and apparent biodegradability of the bioglass coatings in the body, insufficient biomechanical knowledge of endosteal direct anchorage of prosthetic devices is the main reason for failure in these experiments. An exclusive biochemical way of anchoring prosthetic parts in the marrow cavity is rejected. The results presented are discussed in the light of the recent literature.

Q: SQL Query not getting executing - returns empty set I have a database table with date field whose data type is also date and i want to fetch those recods which lie betwnn two dates. My query is : SELECT * FROM wp_races_entry WHERE date_in >=2012-02-08 && date_in<=2012-02-27 i also tried SELECT * FROM wp_races_entry WHERE date_in BETWEEN 2012-02-08 AND 2012-02-27 i have records in table with date 2012-02-14 but still it return empty value. Please help me guiding what i am missing exactly. A: You need quotes round your dates: SELECT * FROM wp_races_entry WHERE date_in BETWEEN '2012-02-08' AND '2012-02-27' Without the quotes your dates are treated as arithmetic expressions: 2012-02-08 = 2002. The query you posted is equivalent to this: SELECT * FROM wp_races_entry WHERE date_in BETWEEN 2002 AND 1983 A: 2012-02-08 isn't a date, it's an integer calulation that yield the result 2002. This is then implictly cast into a date, with 2002 meaning 2002 days from the base date Instead, use '2012-02-08' which is a string, which is also implicitly cast into a date, but the one you want. SELECT * FROM wp_races_entry WHERE date_in BETWEEN '2012-02-08' AND '2012-02-27'

package internal import ( "testing" "time" . "github.com/onsi/gomega" ) func TestRetryBackoff(t *testing.T) { RegisterTestingT(t) for i := -1; i <= 16; i++ { backoff := RetryBackoff(i, time.Millisecond, 512*time.Millisecond) Expect(backoff >= 0).To(BeTrue()) Expect(backoff <= 512*time.Millisecond).To(BeTrue()) } }

In the fabrication of integrated circuits, a number of well-established processes involve the application of ion beams to semiconductor wafers in vacuum. These processes include ion implantation, ion beam milling and reactive ion etching. In each instance, a beam of ions is generated in a source and is directed with varying degrees of acceleration toward a target wafer. Ion implantation has become a standard technique for introducing impurities into semiconductor wafers. The energetic ions in an ion beam applied to a semiconductor wafer generate heat in the wafer. This heat can become significant, depending on the energy level and current level of the ion beam and can result in uncontrolled diffusion of impurities beyond prescribed limits in the wafer. A more severe problem with heating is the degradation of patterned photoresist layers which are often applied to semiconductor wafers before processing and which have relatively low melting points. Other semiconductor wafer processes such as ion etching, sputter deposition and etching, ion beam deposition, vacuum evaporation, plasma etching and chemical vapor deposition are performed in vacuum and may result in undesired heating of the wafer. In some instances, the process may require heat to be transferred to the wafer. In commercial semiconductor processing, a major objective is to achieve a high throughput in terms of wafers processed per unit time. One way to achieve high throughput is by automation of the process for increased speed, reduced human handling of wafers, and more uniform and particulate-free devices. Another way to achieve high throughput in the case of an ion beam system is to use a relatively high current beam so that the desired process is completed in a shorter time. However, large amounts of heat are likely to be generated in the wafers being processed. Thus, it is necessary to provide wafer cooling in order to prevent the above-described detrimental effects of elevated temperatures. A serious difficulty in effecting thermal transfer in vacuum is that thermal energy is blocked from conduction through a vacuum. The rate of thermal transfer from a semiconductor wafer by radiation is inadequate for most processes. Therefore, to achieve adequate rates of thermal transfer by conduction, it is necessary to physically contact the wafer with a thermally conductive medium coupled to a heat sink. Although such a system is straightforward in principle, efficient wafer cooling for automated ion implantation systems has been difficult to achieve for a number of reasons. Since the front of the wafer must be exposed for ion beam treatment, any clamping must be at the periphery or by centrifugal force. Direct solid-to solid contact between a wafer and a flat metal heat sink is relatively ineffective since any bowing or irregularities in the back surface of the wafer result in regions where the wafer does not contact the heat sink surface Furthermore, where contact does occur, microscopic voids in the wafer and heat sink surfaces result in actual physical contact occurring over only about five percent of the two surfaces. As a result, thermal transfer is poor. A contoured heat sink surface for optimizing conductive heat transfer between a wafer and a heat sink is disclosed in U.S. Pat. No. 4,535,835, issued Aug. 20, 1985 to Holden. The heat sink surface is contoured so as to impose a load that results in a uniform contact pressure distribution and a stress approaching the elastic limit of the wafer for a peripherally clamped wafer. Other techniques for limiting wafer temperature during processing have included batch processing in which the incident ion beam is time-shared over a number of wafers so that the heating on any particular wafer is limited. A thermally-conductive fluid can be confined by a flexible diaphragm which contacts the back of the wafer as disclosed in U.S. Pat. Nos. 4,580,619 issued Apr. 8, 1986 to Aitken and 4,682,566 issued July 28, 1987 to Aitken. The technique of gas conduction has also been utilized for wafer cooling in vacuum. Gas is introduced into a cavity behind a semiconductor wafer and effects thermal coupling between the wafer and a heat sink. Gas-assisted solid-to-solid thermal transfer with a semiconductor wafer is disclosed in U S. Pat. No. 4,457,359 issued July 3, 1984 to Holden. A semiconductor wafer is clamped at its periphery onto a shaped platen. Gas under pressure is introduced into the microscopic void region between the platen and the wafer. The gas pressure approaches that of the preloading clamping pressure without any appreciable increase in the wafer to platen spacing, thereby reducing the thermal resistance. When gas conduction cooling is utilized, it is necessary to confine the gas to the region behind the wafer and to prevent escape of the gas into the vacuum chamber, since gas escaping into the vacuum chamber is likely to have detrimental effects on the process being performed. Another prior art technique for thermal transfer in vacuum involves the use of a thermally-conductive polymer between a semiconductor wafer and a heat sink. A tacky, inert polymer film for providing thermal contact between a wafer and a heat sink is disclosed in U.S. Pat. No. 4,139,051 issued Feb. 13, 1979 to Jones et al. The polymer film disclosed by Jones et al has a sticky surface which is used to advantage to retain the wafer in position during processing. However, such a sticky surface is unacceptable in automated processing, wherein the wafer must easily be removed after ion beam treatment. The use of sticky surfaces in automated equipment often results in wafer breakage during wafer removal, or in an inability to remove the wafer from the sticky surface at all. Furthermore, particles, dust and other undesired materials tend to adhere to the sticky polymer surface and to contaminate subsequent wafers. In addition, cleaning of foreign matter from the sticky surface is difficult. An automated wafer clamping mechanism utilizing a pliable thermally-conductive layer between a semiconductor wafer and a heat sink is disclosed in U.S. Pat. No. 4,282,924 issued Aug. 11, 1981 to Faretra. The wafer is clamped at its periphery to a convexly-curved platen having a layer of thermally-conductive silicone rubber on its surface. The Faretra apparatus has provided satisfactory thermal transfer under a variety of conditions. However, sticking of wafers to the silicone rubber surface has occasionally been a problem. To limit such sticking, relatively hard silicone rubbers have been utilized. However, the relatively hard silicone rubber is less effective with respect to thermal transfer, and intimate contact between the wafer and the convexly curved silicone rubber surface is not always achieved. A technique for modifying the surface of a polymeric material utilizing ion implantation of selected ions is disclosed in British Pat. Application No. 2,071,673A, published Sept. 23, 1981. However, the British publication contains no disclosure of a technique for preventing stickiness on polymer surfaces. Imperfections and gas bubbles in the silicone rubber or other polymer layer can seriously degrade thermal transfer performance. When a gas bubble that is present during the molding process leaves a void on the surface of the silicone rubber layer, thermal transfer is reduced in the area of the void. When gas bubbles are located within the bulk of the silicone rubber layer, they can gradually outgas during vacuum processing, thereby causing a virtual leak in the vacuum chamber It has been difficult to achieve a silicone rubber layer that is uniform and free of gas bubbles. While the curved platens disclosed in U.S. Pat. Nos. 4,535,835 and 4,282,924 increase the area of contact between the wafer and the thermally-conductive surface, they introduce a spatial variation in angle between the incident ion beam and the wafer surface. In some processes such as ion implantation, angle-of-incidence variations can be a serious problem. The depth of penetration of incident ions is a function of incident angle because of the well-known channeling effect. Therefore, it is desirable in ion implantation to provide a constant angle-of-incidence between the ion beam and the wafer surface over the surface area of the semiconductor wafer. It is a general object of the present invention to provide improved methods and apparatus for fabricating a polymer layer. It is another object of the present invention to provide methods and apparatus for molding a polymer layer that is substantially free of gas bubbles and surface cavities. It is a further object of the present invention to provide methods and apparatus for molding a polymer layer having a very smooth surface. It is yet another object of the present invention to provide methods and apparatus for molding a polymer layer having high purity. It is another object of the present invention to provide methods and apparatus for molding a thin polymer layer having precisely-controlled dimensions. It is a further object of the present invention to provide a method for fabricating apparatus for thermal transfer with a semiconductor wafer in vacuum. It is still another object of the present invention to provide methods and apparatus for molding a polymer layer, which are simple in operation and low in cost.

LEARN ENCAUSTIC WAX PAINTING ONE ON ONE OR WITH A SMALL GROUP… LEARN AT YOUR OWN PACE. Penny will teach you the ins and outs of painting with encaustic wax. Learn to melt, color and paint with molten beeswax. Burn patterns and depth into with a blow torch or learn to iron the wax on watercolor paper. Choose a style and learn in a personalized one on one setting or in a small party group with friends.

Runs a large IVF programme of at least 600 cases per year to benefit from the economy of scale cost reduction and accumulation of experience An excellent vitirfication (freezing) programme as we now recognize that success rates in the thaw cycle is as good if not better than the fresh cycle. Also the centre should be able to transfer fewer embryos knowing that the frozen embryos do just as well! Sunfert fulfils all the above! At Sunfert, we breathe and live fertility! Start Your Journey To Conception Today! Sign-Up Newsletter CREATIONS, Sunfert’s newsletter will be given a new look, and is packed with more information. Look out for our next issue which will be published soon!

攝大乘論Mahāyāna-saṃparigraha-śāstra, a collection of Mahāyāna śāstras, ascribed to Asaṅga, of which three tr. were made into Chinese. 攝心 To collect the mind, concentrate the attention. 攝念山林 The hill-grove for concentrating the thoughts, a monastery. 攝意音樂 Music that calms the mind, or helps to concentration. 攝拖苾馱Śabda-vidyā, (a śāstra on) grammar, logic. 攝摩騰 Kāśyapa-Mātaṇga, v. 迦 according to tradition the first official Indian monk (along with Gobharana) to arrive in China, circa A.D. 67; tr. the Sūtra of the Forty-two Sections. 攝衆生戒 接生戒 The commands which include or confer blessing on all the living. 攝論 The collected śāstras, v. supra. 攝論宗 The school of the collected śāstras. 曩 Of old, ancient; translit. na. 曩莫nāmaḥ, v. 南. 欄 A rail, handrail; pen, fold. 欄楯 Barrier, railing. 灌 To water, sprinkle, pour; to flow together, or into, accumulate. 灌佛 浴佛 To wash a Buddha's image with scented water, which is a work of great merit and done with much ceremony. 灌室 The building in which the esoterics practise the rite of baptism. 灌洗 To wash a Buddha's image. 灌臘 The washing of a Buddha's image at the end of the monastic year, the end of summer. 灌頂abhiṣecana; mūrdhābhiṣikta; inauguration or consecration by sprinkling, or pouring water on the head; an Indian custom on the investiture of a king, whose head was baptized with water from the four seas and from the rivers in his domain; in China it is administered as a Buddhist rite chiefly to high personages, and for ordination purposes. Amongst the esoterics it is a rite especially administered to their disciples; and they have several categories of baptism, e.g. that of ordinary disciples, of teacher, or preacher, of leader, of office-bearer; also for special causes such as relief from calamity, preparation for the next life, etc. 灌頂住 The tenth stage of a bodhisattva when he is anointed by the Buddhas as a Buddha. 爛 Glittering, as iridescent fish. 爛魚 Rotten, soft; pulp. 瓔 A gem, a necklace. 瓔珞 A necklace of precious stones; things strung together. 竈 A kitchen-stove. 竈神 The kitchen-stove god, or kitchen-god who at the end of each year is supposed to report above on the conduct of members of the family. 纏 To bind with cords; bonds; another name for 煩惱 the passions and delusions, etc. 纏報 The retribution of transmigrational-bondage. 纏無明 The bondage of unenlightenment. 纏縛 Bondage; to bind; also the 十纏 and 四縛 q.v. 續 To join on; continue, add, supplementary, a supplement. 續命 (Prayers for) continued life, for which the 續命神幡 flag of five colours is displayed. 羼 Crowding sheep, confusion; translit. kṣan, ṣan. 羼底 (or羼提) kṣānti, patience, forbearance, enduring shame, one of the six pāramitās. 羼提仙人 Kṣāntiṛṣi, name of Śākyamuni in a previous incarnation, the patient or enduring ṛṣi. 蘭盆 (蘭會) Ullambana, Lambana, Avalamba, v. 盂. The festival of masses for destitute ghosts on the 15th of the 7th month. 蘭菊 Orchid and chrysanthemum, spring and autumn, emblems of beauty. 蘭闍 蘭奢 (蘭奢待) A Mongol or Turkish word implying praise. 蘭香 Orchid fragrance, spring. 蘖 A shrub, tree stump, etc., translit. g, ga, gan. 蘖哩訶 蘖羅訶 Gṛha; Grāha; the seizer, name of a demon. 蘖喇婆garbha, tr. 中心; the womb, interior part. 蘖嚕拏 v. 迦 garuḍa. 蘖馱矩吒 Gandhakutī, a temple for offering incense in the Jetavana monastery and elsewhere). 蠟 Wax. 蠟印 To seal with wax, a wax seal. 覽 To look at, view; translit. raṃ-; associated with fire. 護 To protect, guard, succour. 護世者 The four lokapālas, each protecting one of the four quarters of space, the guardians of the world and of the Buddhist faith. 護命 Protection of life. 護國 The four lokapālas, or rāṣṭrapālas, who protect a country. 護寺vihārapāla, guardian deity of a monastery. 護念 To guard and care for, protect and keep in mind. 護戒神 The five guardian-spirits of each of the five commandments, cf. 二十五神. 護摩homa, also 護磨; 呼麽 described as originally a burnt offering to Heaven; the esoterics adopted the idea of worshipping with fire, symbolizing wisdom as fire burning up the faggots of passion and illusion; and therewith preparing nirvāṇa as food, etc.; cf. 大日經; four kinds of braziers are used, round, semi-circular, square, and octagonal; four, five, or six purposes are recorded i.e. śāntika, to end calamities; pauṣṭika (or puṣṭikarman) for prosperity; vaśīkaraṇa, 'dominating,' intp. as calling down the good by means of enchantments; abhicaraka, exorcising the evil; a fifth is to obtain the loving protection of the Buddhas and bodhisattvas; a sixth divides puṣṭikarman into two parts, the second part being length of life; each of these six has its controlling Buddha and bodhisattvas, and different forms and accessories of worship. 護明大士 Prabhāpāla; guardian of light, or illumination, name of Śākyamuni when in the Tuṣita heaven before earthly incarnation. 護法 To protect or maintain the Buddha-truth; also name of Dharmapāla q.v. 護法神 The four lokapālas, seen at the entrance to Buddhist temples, v, supra. 護童子法 Method of protecting the young against the fifteen evil spirits which seek to harm them. 護符 A charm used by the esoterics. 護苾那 Hupian, 'the capital of Vridjisthāna, probably in the neighbourhood of the present Charekoor... to the north of Cabool.' Eitel. 護身 Protection of the body, for which the charm 護符 is used, and also other methods. 辯 To discuss, argue, discourse. 辯才 Ability to discuss, debate, discourse; rhetoric. 辯才天 Sarasvatī, goddess of speech and learning, v. 大辯才天. 辯無礙 Power of unhindered discourse, perfect freedom of speech or debate, a bodhisattva power. 鑁 Translit. vaṃ, associated with water and the ocean; also, the embodiment of wisdom. 鐶 A metal ring; a ring. 鐶釧 Finger-rings and armlets. 鐵 Iron. 鐵圍山 Cakravāla, Cakravāda. The iron enclosing mountains supposed to encircle the earth, forming the periphery of a world. Mount Meru is the centre and between it and the Iron mountains are the seven 金山 metal-mountains and the eight seas. 鐵城 The iron city, hell. 鐵札 Iron tablets in Hades, on which are recorded each person's crimes and merits. 鐵輪 The iron wheel; also 鐵圍山 Cakravāla, supra. 鐵輪王 Iron-wheel king, ruler of the south and of Jambudvīpa, one of the 四輪王. 露牛 The great white ox and oxcart revealed in the open, i.e. the Mahāyāna, v. Lotus Sūtra. 霹 Crash, rumble. 霹靂 A thunder-crash. 饑 Hunger, famine. 饑餓地獄 The hell of hunger. 饑饉災 The calamity of famine. 饒 Spare; abundance, surplus; to pardon. 饒王 (饒王佛) Lokeśvara, 'the lord or ruler of the world; N. of a Buddha' (M.W.); probably a development of the idea of Brahmā, Viṣṇu or Śiva as lokanātha, 'lord of worlds.' In Indo-China especially it refers to Avalokiteśvara, whose image or face, in masculine form, is frequently seen, e.g. at Angkor. Also 世饒王佛. It is to Lokeśvara that Amitābha announces his forty-eight vows. 饒益 To enrich. 饒舌 A fluent tongue; loquacious. 驅 To drive out or away, expel, urge. 驅烏 Scarecrow, term for an acolyte of from seven to thirteen years of age, he being old enough to drive away crows. 驅龍 Dragon-expeller, a term for an arhat of high character and powers, who can drive away evil nāgas. 髏kapāla; a skull. 髏鬘 A chaplet or wreath of skulls, worn by the Kāpālikas, a Śivaitic sect; kapālī is an epithet of Śiva as the skull-wearer. 鬘 A head-dress, coiffure; a chaplet, wreath, etc.; idem 末利. 魑 A mountain demon resembling a tiger; 魅 is a demon of marshes having the head of a pig and body of a man. The two words are used together indicating evil spirits. 魔 魔羅 Māra, killing, destroying; 'the Destroyer, Evil One, Devil' (M.W.); explained by murderer, hinderer, disturber, destroyer; he is a deva 'often represented with a hundred arms and riding on an elephant'. Eitel. He sends his daughters, or assumes monstrous forms, or inspires wicked men, to seduce or frighten the saints. He 'resides with legions of subordinates in the heaven Paranirmita Vaśavartin situated on the top of the Kāmadhātu'. Eitel. Earlier form 磨; also v. 波 Pāpīyān. He is also called 他化自在天. There are various categories of māras, e.g. the skandha-māra, passion-māra, etc. 魔事 Māra-deeds, especially in hindering Buddha-truth. 魔天 Māra-deva, the god of lust, sin, and death, cf. Māra. 魔女 The daughters of Māra, who tempt men to their ruin. 魔忍 Māra-servitude, the condition of those who obey Māra. 魔怨 Māra enmity; Māra, the enemy of Buddha. 魔戒 Māra-laws, Māra-rules, i.e. those of monks who seek fame and luxury. 魔梵 Māra and Brahmā; i.e. Māra, lord of the sixth desire-heaven, and Brahmā, lord of the heavens of form. 魔檀 Māra-gifts, in contrast with those of Buddha. 魔民mārakāyikas, also 魔子魔女 Māra's people, or subjects. 魔旬 (魔波旬) Māra-pāpīyān, cf. 波. 魔王 The king of māras, the lord of the sixth heaven of the desire-realm.

Hawking letter reveals friendship with UVic professor Every year, UVic Libraries receives approximately 5,000 gifts-in-kind items from the community, each one assessed based on strict criteria that they should support the teaching, research, and learning activities of the Libraries. Twenty-five per cent end up in the circulating collection, with some items going to Special Collections & University Archives owing to their rarity and outstanding provenance, or association with a local author or artist. Gifts-in-kind Coordinator Shelley Coulombe processes a wide range of contributions that include language books, popular fiction, and business related topics. In addition, she says the library is interested in receiving books related to the world wars, Canadian history and literature (especially BC), Greek & Roman studies, education and children’s books for the Curriculum collection, and fine arts. While not all items are selected for use in the library, Coulombe also sorts through an assortment of forgotten ephemera­­­­­ that includes airline boarding passes, bank statements, and even credit card bills. In particular, what catches her eye are the mainstays that donors inserted long ago while enjoying a good read, namely fanciful bookmarks, trendy postcards, family archival photographs, and even foreign currency (most recently, for the British Armed Forces canteen). “What surprises me most is what ends up in the books,” said Coulombe. Recently, one gift stood out that was donated alongside early first-edition Cambridge volumes about black holes, quantum physics and subject matter related to relativity. Engaging a new generation of students at UVic When deciding to donate a portion of his library to UVic Libraries, Professor Emeritus Werner Israel included a first-edition copy of The large scale structure of space-time (Cambridge, 1973) by S.W. Hawking and G.F.R. Ellis. “While some of the textbooks I donated are now over 30 years old, they contain basic material that will never be outdated. It is good to know that they will again see active use by the new generation at UVic,” explains Israel, who retired in 2011 from the Department of Physics. However, when examining the newly donated book, library staff discovered a signed letter from “Stephen” to Werner (February 6, 1981). Typed, the letter was tucked into the back dust jacket, along with two newspaper clippings about Hawking’s latest accomplishment, and Israel’s pencilled summary of important formulae on the flyleaf. Informal and chatty in tone, the letter, likely long forgotten since it was received, reveals a close friendship between kindred spirits and accomplished physicists who collaborated on two titles, General Relativity: An Einstein Centenary Survey (Cambridge, 1979) and Three Hundred Years of Gravitation (Cambridge, 1987). “Editing these books did not involve active research,” Israel said. “So there was no need for daily consultation after the initial planning of topics and authors. I undertook most of the routine work, which would have been a burden for Stephen. But it was, of course, Stephen’s name and charisma which brought ready acceptances from all of the distinguished authors we invited.” An invitation leads to a half-century friendship When Israel completed his PhD from Trinity College, Dublin in 1960, he could not have imagined his future, spreading the gospel of his mentor, the great Irish mathematician John Lighton Synge, and collaborating with Hawking, best-selling author of A Brief History of Time. In 1967, Israel caused general surprise when he presented a theorem indicating that black holes must actually be very simple objects, differing from each other only in their mass and spin. Confirmed and extended by Hawking (among others) in the early 1970s, Israel’s theorem that “black holes have no hair” still forms the basis for all research on the subject today. Their shared interest in black holes led Kip Thorne (co-winner of the 2017 Nobel Prize in Physics) to invite Hawking and Israel to spend the 1974-5 term at Caltech, thus deepening a 1972 friendship between the Israels and the Hawkings, cemented further when the two couples spent the following term together at Stephen’s college in Cambridge. “Notwithstanding his many problems, Stephen had the great kindness to take out our son Mark, to see the first Star Wars film, while Mark was alone in England, living in a Cambridge boarding school. And many years later, when Stephen learned that Mark was confined to a wheelchair and, like himself, unable to speak, he lost no time in writing to us with suggestions of how we might help Mark,” said Israel’s wife Inge. Described as self-deprecating, enthusiastic, obsessive, and absent-minded with a wry sense of humour in a 2011 interview, Israel’s personality complemented Hawking’s overabundant curiosity, generous spirit, and playful nature. Throughout their half-century friendship, Hawking and Israel stayed in touch. In 1991, Hawking came to Banff to attend a conference in celebration of Israel’s 60th birthday, and delivered a lecture to a packed auditorium in Edmonton. Israel acknowledges that it was a special honour to be nominated for Fellowship of the Royal Society (FRS) by Roger Penrose and Hawking, the two leading gravitational theorists of his generation. “On the day of my induction ceremony at the Royal Society headquarters in Burlington House, London (in 1986), Jane (Hawking) drove Stephen and me down from Cambridge, taking us as far as Piccadilly Circus,” Israel said. “From there we were on our own. I never found out why Stephen’s wheelchair always went at almost twice the walking speed, whether it was because it didn’t have a low gear or because of his usual sense of mischief. In Cambridge this was less of a problem, but on the busy sidewalks of London he was a menace.” “Watch out for the wheelchair” In her memoirs, Finding the Words, Inge says “Once Stephen and his wheelchair were deposited on the sidewalk, there was no holding him back. He chased along at such speed that Werner felt obliged to run ahead of him, waving his arms wildly and calling, 'Watch out for the wheelchair’.” "We have wonderful memories of delicious lunches at the Hawking’s home on West Road, prepared single-handedly by Jane. At the table, after serving everyone, she would feed Stephen as if it were the most natural thing in the world, while carrying on a perfectly normal conversation. Then followed a game of croquet on the lawn or an afternoon of music, when Jane would sing and her father accompany her on the piano,” Inge writes. “I still see Stephen whizzing round the garden, giving the children rides on his wheelchair which he maneuvered with skill and great care for the children’s sake.“ Following Hawking’s passing in March 2018, receiving the gift of his letter and book at UVic Libraries is timely and poignant. “We feel very privileged to have known Stephen and especially to have numbered among his friends. He was kind and incredibly brave. I was especially touched at his making the effort to come to my 80th surprise birthday party, organized by a mutual friend in Cambridge in 2007,” says Inge. Today, the 87-year-old retired UVic professor enjoys volunteering and working with students in the ESL (English as Second Language) department, listening to his classical music collection, and enjoying life with his dear wife. Find out more This very special edition can be viewed in Special Collections (QC173.59 S65H38), where readers can experience another dimension of the prolific genius of Stephen Hawking; and by holding a first-edition, signed volume in their hands at UVic Libraries.

Q: How do I turn a SQL database downloaded table into a Google Sheets multi-line chart? I want to turn this table (comes from an SQL database) ...into a line chart in Google Sheets that looks like this A: first you will need to pivot out series with QUERY formula and then plot the chart =QUERY(A1:C11, "select A,C,sum(C) where A is not null group by A,C pivot B", 1)

American Patriots Association The American Patriots Association is a private association of concerned patriotic citizens who share the common bond of believing that the United States of America really is the greatest country on earth. The APA is dedicated to help preserve and protect freedom in America by working to raise awareness and educate people about freedom, democracy and the American way of life.

[Experiences with Wagner's cup prosthesis]. Surface replacement is an alternative procedure especially in younger patients who absolutely need hip replacement. Accurate diagnosis is imperative and the range of indications is naturally small. A follow up study of 44 patients (range, 10 months to 4 years) showed 2 loosenings (5%); other severe complications were not seen. In 39 (89%) hips was no more pain after operation and there was a distinct improvement of motility in 39 (89%) cases. Avascular necrosis should not be treated by surface replacement. Primary inflammatory diseases (rheumatoid arthritis) seem to have a higher incidence of complications, especially when long time treated with large doses of steroids. Cortisone medication promotes avascular necrosis of the hip and lytic enzymes coming from the pannus might be another reason for loosening. According to our experience the position of the cup according to varus or valgus has no influence on the result of the operation. The Watson-Jones approach seems to be better than the approach recommended by Wagner, because paraarticular calcifications are seldom seen.

INTRODUCTION ============ Anorectal melanoma is associated with an extremely poor prognosis regardless of the aggressiveness of surgical therapy; it is commonly incurable at presentation, with many patients developing systemic metastasis within a year after diagnosis.^[@B1]--[@B2]^ However, reports have noted that long-term survival was seen only in patients who underwent an abdominoperineal resection instead of wide local excision.^[@B3]--[@B4]^ It is estimated that only 0.1-0.5% of all rectal tumors are leiomyosarcomas; only 136 cases of rectal leiomyosarcoma had been recorded in a 1986 literature review.^[@B5]^ Local excision carries an approximately 80-85% chance of recurrence.^[@B6]^ Since 80% of these tumors are in the distal rectum, abdominoperineal excision has been the most frequently performed operation. The five-year survival rate in most series seems to be 20-25% after radical surgery.^[@B6]--[@B7]^ Kaposi\'s sarcoma is the most common malignant tumor in AIDS patients.^[@B8]^ Kaposi\'s sarcoma is often asymptomatic and as such usually requires no treatment, but surgery is sometimes indicated to control bleeding or obstructive symptoms. There are still considerable oncologic objections to laparoscopic procedures for cure of colorectal carcinoma. Proponents of laparoscopic abdominoperineal resection cite arguments in favor, that the extended lymph node dissection, the mobilization of the rectum and the mesorectum and stoma creation can be laparoscopically performed; furthermore, the transperineal tumor removal is affected. Nonetheless, local wound recurrence of tumor cells in the port sites of patients who have undergone curative laparoscopic procedures for cancer are of major concern. Due to relative prevalence rates, other series have concentrated on resection of carcinoma. Therefore, the aim of this study was to assess the results of laparoscopic abdominoperineal resection for treatment of non-carcinomatous malignancies. METHODS ======= All five patients were referred to our department for treatment of biopsy-proven neoplasms. In four cases prior local excision had been followed by recurrence. Preoperative staging included computerized axial tomography (CAT) scan and anorectal ultrasound and in all cases failed to identify distant or nodal disease, respectively. All patients underwent conventional preoperative mechanical bowel preparation and received routine oral and parenteral antibiotic prophylaxis. The laparoscopic operative technique for abdominoperineal resection of the rectum has been previously described.^[@B9],[@B10]^ Operative steps include: 1) mobilization of the left colon; 2) division of the inferior mesenteric vessels; 3) division of the mesentery; 4) total mesorectal excision; 5) division of bowel at the sigmoid-descending junction; 6) perineal dissection in the standard fashion with specimen removal; and 7) end colostomy creation. The operative time, intraoperative findings, transfusion requirement, intra- and postoperative complications, length of hospitalization, and outcomes of surgery were recorded for each patient. REPORT OF CASES AND RESULTS =========================== Case 1. ------- A 75-year-old female patient presented to another surgeon with a primary tumor located 5 cm cephalad to the dentate line, on the posterior lateral aspect of the rectum. Macroscopically the tumor was ulcerated and exophytic. Histopathologic evaluation revealed an invasive malignant melanoma with vascular involvement; high mitotic activity (mean 6 mitoses per 107 high-power field \[HPF\]); and a positive immunocytochemical profile of 100S-HMB 45/ 50. Chest, abdominal and pelvic CT scans failed to reveal any local or distant metastasis. The melanoma was transanally removed. Two months later a biopsy failed to identify any local recurrence. Six months after the local excision the patient presented with a gelatinous 4 cm diameter mass, localized on the left postero-lateral aspect of the dentate line, extending to the anorectal ring. A pelvic and abdominal CT scan revealed a dense 2 cm ovoid mass on the left lateral wall of rectum, caudal to the levator ani, involving the puborectalis muscle. Although the patient had enlarged lymph nodes in the left ischiorectal fossa at the level of the levator ani, there were no distant metastases noted. At that time the patient was referred to our department for evaluation of severe pain and rectal bleeding and underwent a laparoscopic abdominoperineal resection. Pathologic assessment showed metastatic, multicentric melanoma of the rectum, involving 8 of 9 lymph nodes. The caudal lesion was 1.2 cm from the distal margin of the specimen. The patient had an uneventful recovery and was discharged on postoperative day seven. She declined any adjuvant therapy. Four months after abdominoperineal resection, the CT scan found the liver extensively involved with metastatic lesions without port-site or local metastasis. Case 2. ------- An 87-year-old male patient presented to another surgeon with a malignant melanoma infiltrating the external anal sphincter and puborectalis muscle. A wide local excision of the lesion was performed 3 cm cephalad to the dentate line. The patient was followed up for 18 months, during which time no distant or local metastases were revealed. Eighteen months later he was referred to our department with a deeply infiltrative 3 cm diameter anteriorly bound tumor at the level of the dentate line, involving the anterior rectal wall. He complained of tenesmus and rectal bleeding. Rectal ultrasound revealed invasion of both the internal and external anal sphincters. After abdominoperineal resection, the pathologist reported a tumor free distal margin of 0.8 cm, and 3 of the 7 pericolic harvested lymph nodes had metastatic foci. Postoperatively he developed a common bile duct obstruction due to stone impaction at the ampulla of Vater, which was successfully removed by endoscopic retrograde cholangiopancreaticography. No adjuvant therapy was administered. The patient had uneventful recovery and two months after surgery had no evidence of any distal, local or port-site recurrence. Case 3. ------- A 69-year-old patient presented to another surgeon with a broad-based polypoid lesion of 1.5 cm diameter in the anal canal removed by endoscopic snare. Histologic studies showed a poorly differentiated primary tumor, with high mitotic activity which infiltrated the rectal wall; immunohistochemical profile was positive for S100-HMB 45/50. The patient was followed up for two months, after which a biopsy revealed a local recurrence of malignant melanoma at the anterior dentate line. The patient was referred to our department for curative resection of an early recurrence. Rectal ultrasonography and anoscopic evaluation revealed a 3 mm diameter anteriorly based lesion at the dentate line. Because preoperative CT did not reveal any distant metastasis, the patient underwent laparoscopy with a view towards abdominoperineal resection. Unfortunately, two 4 cm diameter right-sided hepatic metastasis were identified and confirmed by percutaneous fine needle biopsy. Therefore, the decision was made not to proceed with an abdominoperineal resection. The patient was discharged home on postoperative day two without any complications. She died secondary to liver metastasis but without any port-site recurrence 46 days after surgery. Case 4. ------- A 68-year-old female patient who was receiving immunosuppressive agents for rheumatoid arthritis was referred to our department for a primary tumor presenting as a firm mass in the left wall of the anal canal. A graytan soft tissue weighing 42 grams within a diameter of 5.5 × 3.6 × 3.5 cm was removed by wide local excision, and histologic evaluation confirmed the diagnosis of leiomyosarcoma. After seven months, the patient presented with a recurrent anal mass. CT scan and anal ultrasonography revealed a localized lesion consistent with a possible recurrence. The patient underwent laparoscopic abdominoperineal resection after which the pathologist described the lesion as a lymphosarcoma. The patient was discharged on the fifth postoperative day with no complications. Two years later, the patient presented with a 2 cm diameter left obturator mass, with bilobar hepatic and bilateral pulmonary nodules. However, neither CT scan nor magnetic resonance imaging revealed any port-site recurrences. Case 5. ------- A 34-year-old white male patient presented to another surgeon with an advanced HIV infection. During his course, the patient had experienced both cutaneous and gastrointestinal Kaposi\'s sarcoma, cytomegalovirus gastritis, cryptosporidium, peripheral neuropathy, wasting, neutropenia and anemia. The patient presented with tenesmus and continued bleeding with evacuation. Office evaluation revealed two 4 cm diameter Kaposi\'s lesions in the rectum, one 5 cm and one 8 cm cephalad to the dentate line. A circumanal intracutaneous 10 cm diameter perianal Kaposi\'s sarcoma was also noted. A lesion was found in the transverse colon during colonoscopy. At first the patient was treated by chemotherapy and radiotherapy. However, one month later, he presented with obstructive symptoms, increased bleeding, and severe pain. Proctoscopy revealed near-occlusion of the lumen due to the two rectal lesions. Accordingly, it was elected to proceed with a laparoscopic loop ileostomy after which he was discharged home on postoperative day five in stable condition. The patient died five months after the procedure due to systemic cytomegalovirus infection and seizures. No port-site recurrences were noted at the time of death. DISCUSSION ========== Anal cancer is a relatively infrequent tumor, accounting for less than 2% of all large bowel cancer and less than 2000 cases annually in the United States.^[@B11]--[@B13]^ Malignant melanoma constitutes only 3% of tumors arising in the anal canal.^[@B14]^ Only 1.5% of all malignant melanomas develop in the anorectal region and only 460 cases have been reported in the medical literature.^[@B2]^ Anorectal melanoma is associated with an extremely poor prognosis despite aggressive surgical therapy. In the literature the average survival rates vary from 9 months to 2.8 years for all patients with only a 10% five-year survival.^[@B15]^ Brady et al.^[@B16]^ (Memorial Sloan-Kettering Cancer Center) published 85 cases; seventy-five percent of patients had tumors greater than 1 cm with a mean survival of only 17 months. Only six patients survived five years, and all of these six had undergone abdominoperineal resection. Interestingly, two of the long-term survivors had mesenteric lymph nodes involved by tumor. Several other authors^[@B1]--[@B4]^ have found that long-term survival was seen only in patients who underwent abdominoperineal resection and have strongly suggested this procedure for cure. A Swedish group reported that only 2 of 33 patients treated for cure survived for five years.^[@B17]^ Although one of the two survivors underwent local excision, they found that local recurrences occurred significantly more often after a local excision than after an abdominoperineal resection (50% vs. 27%, respectively). Other researchers believe that radical surgery does not alter the natural history of a high rate of distant metastasis and, therefore, advocate sphincter-sparing surgery. Siegel et al.,^[@B18]^ in a review of 30 patients, found that the only two 5-year survivors had been treated with local excision. Similarly, Cooper et al.^[@B19]^ reported that two of six 5-year survivors had local excision. Although melanoma has been successfully used to produce tumor cell lines with increased metastatic capacity from many other experimental tumors,^[@B20]^ none of the three patients in our small series developed port-site recurrences. Leiomyosarcoma of the anorectum is rare, and wide local excision is the best treatment option. Radiation or chemotherapy, alone or in combination, have not been found to be effective, although they may play a palliative role.^[@B21]^ As emphasized in a St. Mark\'s series,^[@B22]^ the treatment of choice for leiomyosarcoma of the rectum arising from sites other than the muscularis mucosa is abdominoperineal excision. In this current series, one patient who was treated with a previous wide local excision previously had local recurrence and finally underwent laparoscopic abdominoperineal resection. Kaposi\'s sarcoma has been reported from the mouth to the anus. Although most of these lesions are asymptomatic, complications can include bleeding, obstruction, intussusception and mesenteric cyst formation. Treatment of anorectal Kaposi\'s sarcoma rarely entails surgery although neither the chemotherapy, radiotherapy, nor immunotheraphy is of proven benefit.^[@B23]^ However, on occasion, fecal diversion or resection of the lesion is necessary. Because of the decreased immune function in HIV-infected patients, laparoscopy may be beneficial.^[@B24]^ CONCLUSION ========== These lesions described in this small series of case reports are infrequently occurring but highly aggressive. It, therefore, seems logical that if these patients need surgery, they be offered the least traumatic, least immune-system compromising type available. In addition, therapy which can expedite hospital discharge and minimize disability is desirable in patients with a limited life expectancy. Issues germane to care of carcinomatous lesions, such as port-site recurrences, may not be as relevant in these other tumors. This study has been funded in part by a generous grant from Ethicon Endosurgery, Inc., Cincinatti, OH.

Effect of angiotensins on electrogenic anion transport in monolayer cultures of rat epididymis. Confluent monolayers cultured from the rat cauda epididymidis have been shown to respond to angiotensin I (AI) and angiotensin II (AII) when studied under short-circuit conditions and bathed on both sides with Krebs-Henseleit solution. Both the decapeptide AI and the octapeptide AII elicited transient increases in short-circuit current (SCC) when added to the basolateral as well as to the apical surfaces, with the effect of basolateral application greater than that of apical application. The maximal responses produced by AI and AII were similar with median effective concentrations of 20 to 80 nmol/l. The increase in SCC by AII was dependent upon extracellular Cl- and was inhibited by addition of a Cl- channel blocker, diphenylamine 2-carboxylate, to the apical surface. These patterns of activity suggest that the SCC responses to angiotensins result from electrogenic chloride secretion. Pretreating the monolayers with captopril (100 nmol/l), an angiotensin-converting enzyme (ACE) inhibitor, reduced the response to basolateral application of AI, but completely abolished the response to AI added apically. These results suggest that the response to apical addition of AI was due to conversion of AI to AII which interacts with apical angiotensin receptors. This conversion was mediated by ACE which has been detected in epididymal monolayers. Of the endogenous ACE activity, 86% was found to be inhibited by captopril (100 nmol/l). Responses of the epididymal monolayers to angiotensins were mediated by specific angiotensin receptors. [Sar1,Ile8]-AII, a specific antagonist of the AII receptor, completely inhibited the responses to AI and AII but had no effect on the responses to bradykinin and endothelin.(ABSTRACT TRUNCATED AT 250 WORDS)

Montgomery FLORIST - LEE & LAN FLORIST - Roses Show your love, passion, and gratitude with a dozen red roses. Accented with white waxflower, an arrangement this full of love is sure to please! Call our shop to order or order flowers now from our website! Show your love, passion, and gratitude with a dozen red roses. Accented with white waxflower, an arrangement this full of love is sure to please! Call our shop to order or order flowers now from our website! Show your love, passion, and gratitude with a dozen red roses. Accented with white waxflower, an arrangement this full of love is sure to please! Call our shop to order or order flowers now from our website! From love to friendship, bright yellow roses are a promise of bright days that stand the test of time. With hosta leaves and green hypericum berries, this dozen yellow roses is a new twist on a classic favorite. Roses from LEE & LAN FLORIST in Montgomery, AL always make an impression. You can express your self in a variety of ways from the simple rose bud vase, to an extravagant two dozen. Browse our rose pictures to find the best arrangement of roses for you. If you don't see something you like give LEE & LAN FLORIST a call. We can arrange roses in a variety of styles to suit your special occasion, style or color such as: yellow roses, white roses and pink roses. Send your message of love with roses from LEE & LAN FLORIST today.

vrijdag 20 november 2015 Jonathan Pollard is free at last Longtime prisoner Jonathan Pollard has just left the Federal Correction Complex in Butner, North Carolina, after over 30 years of being held on espionage charges, his wife Esther announced shortly after 11 a.m. Friday. Esther and a number of Pollard's closest confidantes were waiting to greet Pollard as he first tasted freedom, at 4:15 a.m. EST; from there they will set off for New York to begin their lives anew. There in New York, a probation officer will be assigned to Pollard by the US Department of Justice, and will inspect to make sure the 61-year-old stays within all of the limitations placed on him. Any excursion beyond the immediate area of his residence will require the approval of the probation officer. He will likewise be forbidden from flying to Israel - and will even be forbidden from checking the internet. Pollard, who was arrested on charges of spying for Israel in 1985 and later sentenced to life in prison, began his 31st year in jail as prisoner 09185-016 this past November. He has been suffering from poor health and has become the subject of a high-profile campaign for his release. Last year, the Parole Board of the Justice Department rejected Pollard's parole, with senior U.S. officials involved in the case writing to US President Barack Obama to complain that the decision was "deeply flawed". Once Pollard's release was granted, however, controversy remained over the conditions of his parole, after Obama and other top-ranking officials refused to intervene to allow him to return home to Israel before the 5-year limit. Israeli Politicians welcomed Pollard's release Friday. "The Jewish people welcome the release of Jonathan Pollard," Prime Minister Binyamin Netanyahu said. "As someone who brought up the issue many years with the US President, I have longed for this day." "After three long and difficult decades, Jonathan will finally, finally be reunited with his family," Netanyahu continued. "I wish Jonathan that this Shabbat give him happiness and quiet, and that it should continue for the rest of his life." 67 Welcome To our Christian-Zionists Pro-Israël Blog in Dutch and English language. We are based in The Netherlands, Europe. 1. Please share & link our blog with others.2. Check out our archive in left column.3. Search articles in left above search-box.4. Send us your relevant news (to our e-mail).5. If you feel to support our work for the Lord, or like to donate - then e-mail us.

/* Copyright 2003-2013 Joaquin M Lopez Munoz. * Distributed under the Boost Software License, Version 1.0. * (See accompanying file LICENSE_1_0.txt or copy at * http://www.boost.org/LICENSE_1_0.txt) * * See http://www.boost.org/libs/multi_index for library home page. */ #ifndef BOOST_MULTI_INDEX_DETAIL_SAFE_MODE_HPP #define BOOST_MULTI_INDEX_DETAIL_SAFE_MODE_HPP #if defined(_MSC_VER) #pragma once #endif /* Safe mode machinery, in the spirit of Cay Hortmann's "Safe STL" * (http://www.horstmann.com/safestl.html). * In this mode, containers of type Container are derived from * safe_container<Container>, and their corresponding iterators * are wrapped with safe_iterator. These classes provide * an internal record of which iterators are at a given moment associated * to a given container, and properly mark the iterators as invalid * when the container gets destroyed. * Iterators are chained in a single attached list, whose header is * kept by the container. More elaborate data structures would yield better * performance, but I decided to keep complexity to a minimum since * speed is not an issue here. * Safe mode iterators automatically check that only proper operations * are performed on them: for instance, an invalid iterator cannot be * dereferenced. Additionally, a set of utilty macros and functions are * provided that serve to implement preconditions and cooperate with * the framework within the container. * Iterators can also be unchecked, i.e. they do not have info about * which container they belong in. This situation arises when the iterator * is restored from a serialization archive: only information on the node * is available, and it is not possible to determine to which container * the iterator is associated to. The only sensible policy is to assume * unchecked iterators are valid, though this can certainly generate false * positive safe mode checks. * This is not a full-fledged safe mode framework, and is only intended * for use within the limits of Boost.MultiIndex. */ /* Assertion macros. These resolve to no-ops if * !defined(BOOST_MULTI_INDEX_ENABLE_SAFE_MODE). */ #if !defined(BOOST_MULTI_INDEX_ENABLE_SAFE_MODE) #undef BOOST_MULTI_INDEX_SAFE_MODE_ASSERT #define BOOST_MULTI_INDEX_SAFE_MODE_ASSERT(expr,error_code) ((void)0) #else #if !defined(BOOST_MULTI_INDEX_SAFE_MODE_ASSERT) #include <boost/assert.hpp> #define BOOST_MULTI_INDEX_SAFE_MODE_ASSERT(expr,error_code) BOOST_ASSERT(expr) #endif #endif #define BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(it) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_valid_iterator(it), \ safe_mode::invalid_iterator); #define BOOST_MULTI_INDEX_CHECK_DEREFERENCEABLE_ITERATOR(it) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_dereferenceable_iterator(it), \ safe_mode::not_dereferenceable_iterator); #define BOOST_MULTI_INDEX_CHECK_INCREMENTABLE_ITERATOR(it) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_incrementable_iterator(it), \ safe_mode::not_incrementable_iterator); #define BOOST_MULTI_INDEX_CHECK_DECREMENTABLE_ITERATOR(it) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_decrementable_iterator(it), \ safe_mode::not_decrementable_iterator); #define BOOST_MULTI_INDEX_CHECK_IS_OWNER(it,cont) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_is_owner(it,cont), \ safe_mode::not_owner); #define BOOST_MULTI_INDEX_CHECK_SAME_OWNER(it0,it1) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_same_owner(it0,it1), \ safe_mode::not_same_owner); #define BOOST_MULTI_INDEX_CHECK_VALID_RANGE(it0,it1) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_valid_range(it0,it1), \ safe_mode::invalid_range); #define BOOST_MULTI_INDEX_CHECK_OUTSIDE_RANGE(it,it0,it1) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_outside_range(it,it0,it1), \ safe_mode::inside_range); #define BOOST_MULTI_INDEX_CHECK_IN_BOUNDS(it,n) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_in_bounds(it,n), \ safe_mode::out_of_bounds); #define BOOST_MULTI_INDEX_CHECK_DIFFERENT_CONTAINER(cont0,cont1) \ BOOST_MULTI_INDEX_SAFE_MODE_ASSERT( \ safe_mode::check_different_container(cont0,cont1), \ safe_mode::same_container); #if defined(BOOST_MULTI_INDEX_ENABLE_SAFE_MODE) #include <boost/config.hpp> /* keep it first to prevent nasty warns in MSVC */ #include <algorithm> #include <boost/detail/iterator.hpp> #include <boost/multi_index/detail/access_specifier.hpp> #include <boost/multi_index/detail/iter_adaptor.hpp> #include <boost/multi_index/safe_mode_errors.hpp> #include <boost/noncopyable.hpp> #if !defined(BOOST_MULTI_INDEX_DISABLE_SERIALIZATION) #include <boost/serialization/split_member.hpp> #include <boost/serialization/version.hpp> #endif #if defined(BOOST_HAS_THREADS) #include <boost/detail/lightweight_mutex.hpp> #endif namespace boost{ namespace multi_index{ namespace safe_mode{ /* Checking routines. Assume the best for unchecked iterators * (i.e. they pass the checking when there is not enough info * to know.) */ template<typename Iterator> inline bool check_valid_iterator(const Iterator& it) { return it.valid()||it.unchecked(); } template<typename Iterator> inline bool check_dereferenceable_iterator(const Iterator& it) { return (it.valid()&&it!=it.owner()->end())||it.unchecked(); } template<typename Iterator> inline bool check_incrementable_iterator(const Iterator& it) { return (it.valid()&&it!=it.owner()->end())||it.unchecked(); } template<typename Iterator> inline bool check_decrementable_iterator(const Iterator& it) { return (it.valid()&&it!=it.owner()->begin())||it.unchecked(); } template<typename Iterator> inline bool check_is_owner( const Iterator& it,const typename Iterator::container_type& cont) { return (it.valid()&&it.owner()==&cont)||it.unchecked(); } template<typename Iterator> inline bool check_same_owner(const Iterator& it0,const Iterator& it1) { return (it0.valid()&&it1.valid()&&it0.owner()==it1.owner())|| it0.unchecked()||it1.unchecked(); } template<typename Iterator> inline bool check_valid_range(const Iterator& it0,const Iterator& it1) { if(!check_same_owner(it0,it1))return false; if(it0.valid()){ Iterator last=it0.owner()->end(); if(it1==last)return true; for(Iterator first=it0;first!=last;++first){ if(first==it1)return true; } return false; } return true; } template<typename Iterator> inline bool check_outside_range( const Iterator& it,const Iterator& it0,const Iterator& it1) { if(!check_same_owner(it0,it1))return false; if(it0.valid()){ Iterator last=it0.owner()->end(); bool found=false; Iterator first=it0; for(;first!=last;++first){ if(first==it1)break; /* crucial that this check goes after previous break */ if(first==it)found=true; } if(first!=it1)return false; return !found; } return true; } template<typename Iterator,typename Difference> inline bool check_in_bounds(const Iterator& it,Difference n) { if(it.unchecked())return true; if(!it.valid()) return false; if(n>0) return it.owner()->end()-it>=n; else return it.owner()->begin()-it<=n; } template<typename Container> inline bool check_different_container( const Container& cont0,const Container& cont1) { return &cont0!=&cont1; } /* Invalidates all iterators equivalent to that given. Safe containers * must call this when deleting elements: the safe mode framework cannot * perform this operation automatically without outside help. */ template<typename Iterator> inline void detach_equivalent_iterators(Iterator& it) { if(it.valid()){ { #if defined(BOOST_HAS_THREADS) boost::detail::lightweight_mutex::scoped_lock lock(it.cont->mutex); #endif Iterator *prev_,*next_; for( prev_=static_cast<Iterator*>(&it.cont->header); (next_=static_cast<Iterator*>(prev_->next))!=0;){ if(next_!=&it&&*next_==it){ prev_->next=next_->next; next_->cont=0; } else prev_=next_; } } it.detach(); } } template<typename Container> class safe_container; /* fwd decl. */ } /* namespace multi_index::safe_mode */ namespace detail{ class safe_container_base; /* fwd decl. */ class safe_iterator_base { public: bool valid()const{return cont!=0;} bool unchecked()const{return unchecked_;} inline void detach(); void uncheck() { detach(); unchecked_=true; } protected: safe_iterator_base():cont(0),next(0),unchecked_(false){} explicit safe_iterator_base(safe_container_base* cont_): unchecked_(false) { attach(cont_); } safe_iterator_base(const safe_iterator_base& it): unchecked_(it.unchecked_) { attach(it.cont); } safe_iterator_base& operator=(const safe_iterator_base& it) { unchecked_=it.unchecked_; safe_container_base* new_cont=it.cont; if(cont!=new_cont){ detach(); attach(new_cont); } return *this; } ~safe_iterator_base() { detach(); } const safe_container_base* owner()const{return cont;} BOOST_MULTI_INDEX_PRIVATE_IF_MEMBER_TEMPLATE_FRIENDS: friend class safe_container_base; #if !defined(BOOST_NO_MEMBER_TEMPLATE_FRIENDS) template<typename> friend class safe_mode::safe_container; template<typename Iterator> friend void safe_mode::detach_equivalent_iterators(Iterator&); #endif inline void attach(safe_container_base* cont_); safe_container_base* cont; safe_iterator_base* next; bool unchecked_; }; class safe_container_base:private noncopyable { public: safe_container_base(){} BOOST_MULTI_INDEX_PROTECTED_IF_MEMBER_TEMPLATE_FRIENDS: friend class safe_iterator_base; #if !defined(BOOST_NO_MEMBER_TEMPLATE_FRIENDS) template<typename Iterator> friend void safe_mode::detach_equivalent_iterators(Iterator&); #endif ~safe_container_base() { /* Detaches all remaining iterators, which by now will * be those pointing to the end of the container. */ for(safe_iterator_base* it=header.next;it;it=it->next)it->cont=0; header.next=0; } void swap(safe_container_base& x) { for(safe_iterator_base* it0=header.next;it0;it0=it0->next)it0->cont=&x; for(safe_iterator_base* it1=x.header.next;it1;it1=it1->next)it1->cont=this; std::swap(header.cont,x.header.cont); std::swap(header.next,x.header.next); } safe_iterator_base header; #if defined(BOOST_HAS_THREADS) boost::detail::lightweight_mutex mutex; #endif }; void safe_iterator_base::attach(safe_container_base* cont_) { cont=cont_; if(cont){ #if defined(BOOST_HAS_THREADS) boost::detail::lightweight_mutex::scoped_lock lock(cont->mutex); #endif next=cont->header.next; cont->header.next=this; } } void safe_iterator_base::detach() { if(cont){ #if defined(BOOST_HAS_THREADS) boost::detail::lightweight_mutex::scoped_lock lock(cont->mutex); #endif safe_iterator_base *prev_,*next_; for(prev_=&cont->header;(next_=prev_->next)!=this;prev_=next_){} prev_->next=next; cont=0; } } } /* namespace multi_index::detail */ namespace safe_mode{ /* In order to enable safe mode on a container: * - The container must derive from safe_container<container_type>, * - iterators must be generated via safe_iterator, which adapts a * preexistent unsafe iterator class. */ template<typename Container> class safe_container; template<typename Iterator,typename Container> class safe_iterator: public detail::iter_adaptor<safe_iterator<Iterator,Container>,Iterator>, public detail::safe_iterator_base { typedef detail::iter_adaptor<safe_iterator,Iterator> super; typedef detail::safe_iterator_base safe_super; public: typedef Container container_type; typedef typename Iterator::reference reference; typedef typename Iterator::difference_type difference_type; safe_iterator(){} explicit safe_iterator(safe_container<container_type>* cont_): safe_super(cont_){} template<typename T0> safe_iterator(const T0& t0,safe_container<container_type>* cont_): super(Iterator(t0)),safe_super(cont_){} template<typename T0,typename T1> safe_iterator( const T0& t0,const T1& t1,safe_container<container_type>* cont_): super(Iterator(t0,t1)),safe_super(cont_){} safe_iterator& operator=(const safe_iterator& x) { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(x); this->base_reference()=x.base_reference(); safe_super::operator=(x); return *this; } const container_type* owner()const { return static_cast<const container_type*>( static_cast<const safe_container<container_type>*>( this->safe_super::owner())); } /* get_node is not to be used by the user */ typedef typename Iterator::node_type node_type; node_type* get_node()const{return this->base_reference().get_node();} private: friend class boost::multi_index::detail::iter_adaptor_access; reference dereference()const { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_DEREFERENCEABLE_ITERATOR(*this); return *(this->base_reference()); } bool equal(const safe_iterator& x)const { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(x); BOOST_MULTI_INDEX_CHECK_SAME_OWNER(*this,x); return this->base_reference()==x.base_reference(); } void increment() { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_INCREMENTABLE_ITERATOR(*this); ++(this->base_reference()); } void decrement() { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_DECREMENTABLE_ITERATOR(*this); --(this->base_reference()); } void advance(difference_type n) { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_IN_BOUNDS(*this,n); this->base_reference()+=n; } difference_type distance_to(const safe_iterator& x)const { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(x); BOOST_MULTI_INDEX_CHECK_SAME_OWNER(*this,x); return x.base_reference()-this->base_reference(); } #if !defined(BOOST_MULTI_INDEX_DISABLE_SERIALIZATION) /* Serialization. Note that Iterator::save and Iterator:load * are assumed to be defined and public: at first sight it seems * like we could have resorted to the public serialization interface * for doing the forwarding to the adapted iterator class: * ar<<base_reference(); * ar>>base_reference(); * but this would cause incompatibilities if a saving * program is in safe mode and the loading program is not, or * viceversa --in safe mode, the archived iterator data is one layer * deeper, this is especially relevant with XML archives. * It'd be nice if Boost.Serialization provided some forwarding * facility for use by adaptor classes. */ friend class boost::serialization::access; BOOST_SERIALIZATION_SPLIT_MEMBER() template<class Archive> void save(Archive& ar,const unsigned int version)const { BOOST_MULTI_INDEX_CHECK_VALID_ITERATOR(*this); this->base_reference().save(ar,version); } template<class Archive> void load(Archive& ar,const unsigned int version) { this->base_reference().load(ar,version); safe_super::uncheck(); } #endif }; template<typename Container> class safe_container:public detail::safe_container_base { typedef detail::safe_container_base super; public: void detach_dereferenceable_iterators() { typedef typename Container::iterator iterator; iterator end_=static_cast<Container*>(this)->end(); iterator *prev_,*next_; for( prev_=static_cast<iterator*>(&this->header); (next_=static_cast<iterator*>(prev_->next))!=0;){ if(*next_!=end_){ prev_->next=next_->next; next_->cont=0; } else prev_=next_; } } void swap(safe_container<Container>& x) { super::swap(x); } }; } /* namespace multi_index::safe_mode */ } /* namespace multi_index */ #if !defined(BOOST_MULTI_INDEX_DISABLE_SERIALIZATION) namespace serialization{ template<typename Iterator,typename Container> struct version< boost::multi_index::safe_mode::safe_iterator<Iterator,Container> > { BOOST_STATIC_CONSTANT( int,value=boost::serialization::version<Iterator>::value); }; } /* namespace serialization */ #endif } /* namespace boost */ #endif /* BOOST_MULTI_INDEX_ENABLE_SAFE_MODE */ #endif

Commercially available abrasive materials, such as sandpaper and the like, are generally produced by adhering an abrasive grit onto a substrate. In the case of sandpaper, a paper substrate is used. When used to sand an area, sandpaper is generally used by affixing the sandpaper onto a form, such as a sanding block, and then sanding the area until the abrasive grit on the sandpaper is either "clogged" by the material removed from the item or is detached from the paper backing due to the force of friction. After either of these occurrences, the abrasive paper must be removed from the form and replaced with new sandpaper, a time consuming process. In situations where a curved, or non-planar, surface requires sanding, e.g., cylindrical chair legs, the use of paper is particularly inadequate as any forms used in the sanding process, for best results, must also be curve-shaped. Accordingly, the flat sheet of sandpaper must be adhered to a curved form. This, however, makes changing paper troublesome and adds to the time required to sand an item. In view of the problems associated with the abrasive surfaces and methods for producing these surfaces described previously, there exists a need for an aerosol composition and related method for providing an abrasive, sandpaper-like surface which may be used almost immediately after its production to remove paint and other coatings from selected items as well as to smooth wood, metal, plaster, and like surfaces. Moreover, a composition and related method are needed which would allow a new layer of abrasive material to be quickly and easily applied to a surface that has been rendered relatively non-abrasive through use, thereby rejuvenating the abrasive surface. Further, there is a specific need for an abrasive surface which is capable of being used to sand a curved item and which may be easily and quickly rejuvenated. These and other advantages of the present invention, as well as additional inventive features, will become apparent from the description which follows.

# -*- coding: utf-8 -*- # Copyright (C) 2010-2012, eskerda <eskerda@gmail.com> # Distributed under the AGPL license, see LICENSE.txt import json try: # Python 2 from HTMLParser import HTMLParser except ImportError: # Python 3 from html.parser import HTMLParser from lxml import etree from .base import BikeShareSystem, BikeShareStation from . import utils __all__ = ['Cyclocity','CyclocityStation','CyclocityWeb','CyclocityWebStation'] api_root = "https://api.jcdecaux.com/vls/v1/" endpoints = { 'contracts': 'contracts?apiKey={api_key}', 'stations' : 'stations?apiKey={api_key}&contract={contract}', 'station' : 'stations/{station_id}?contract={contract}&apiKey={api_key}' } html_parser = HTMLParser() class Cyclocity(BikeShareSystem): sync = True authed = True meta = { 'system': 'Cyclocity', 'company': ['JCDecaux'], 'license': { 'name': 'Open Licence', 'url': 'https://developer.jcdecaux.com/#/opendata/licence' }, 'source': 'https://developer.jcdecaux.com' } def __init__(self, tag, meta, contract, key): super( Cyclocity, self).__init__(tag, meta) self.contract = contract self.api_key = key self.stations_url = api_root + endpoints['stations'].format( api_key = self.api_key, contract = contract ) self.station_url = api_root + endpoints['station'].format( api_key = self.api_key, contract = contract, station_id = '{station_id}' ) def update(self, scraper = None): if scraper is None: scraper = utils.PyBikesScraper() data = json.loads(scraper.request(self.stations_url)) stations = [] for info in data: try: station = CyclocityStation(info, self.station_url) except Exception: continue stations.append(station) self.stations = stations @staticmethod def get_contracts(api_key, scraper = None): if scraper is None: scraper = utils.PyBikesScraper() url = api_root + endpoints['contracts'].format(api_key = api_key) return json.loads(scraper.request(url)) class CyclocityStation(BikeShareStation): def __init__(self, jcd_data, station_url): super(CyclocityStation, self).__init__() self.name = jcd_data['name'] self.latitude = jcd_data['position']['lat'] self.longitude = jcd_data['position']['lng'] self.bikes = jcd_data['available_bikes'] self.free = jcd_data['available_bike_stands'] self.extra = { 'uid': jcd_data['number'], 'address': jcd_data['address'], 'status': jcd_data['status'], 'banking': jcd_data['banking'], 'bonus': jcd_data['bonus'], 'last_update': jcd_data['last_update'], 'slots': jcd_data['bike_stands'] } self.url = station_url.format(station_id = jcd_data['number']) if self.latitude is None or self.longitude is None: raise Exception('An station needs a lat/lng to be defined!') def update(self, scraper = None, net_update = False): if scraper is None: scraper = utils.PyBikesScraper() super(CyclocityStation, self).update() if net_update: status = json.loads(scraper.request(self.url)) self.__init__(status, self.url) return self class CyclocityWeb(BikeShareSystem): sync = False meta = { 'system': 'Cyclocity', 'company': ['JCDecaux'] } _list_url = '/service/carto' _station_url = '/service/stationdetails/{city}/{id}' def __init__(self, tag, meta, endpoint, city): super(CyclocityWeb, self).__init__(tag, meta) self.endpoint = endpoint self.city = city self.list_url = endpoint + CyclocityWeb._list_url self.station_url = endpoint + CyclocityWeb._station_url def update(self, scraper = None): if scraper is None: scraper = utils.PyBikesScraper() xml_markers = scraper.request(self.list_url) dom = etree.fromstring(xml_markers.encode('utf-7')) markers = dom.xpath('/carto/markers/marker') stations = [] for marker in markers: station = CyclocityWebStation.from_xml(marker) station.url = self.station_url.format( city = self.city, id = station.extra['uid'] ) stations.append(station) self.stations = stations class CyclocityWebStation(BikeShareStation): @staticmethod def from_xml(marker): station = CyclocityWebStation() station.name = marker.get('name').title() station.latitude = float(marker.get('lat')) station.longitude = float(marker.get('lng')) station.extra = { 'uid': int(marker.get('number')), 'address': html_parser.unescape( marker.get('fullAddress').rstrip() ), 'open': int(marker.get('open')) == 1, 'bonus': int(marker.get('bonus')) == 1 } return station def update(self, scraper = None): if scraper is None: scraper = utils.PyBikesScraper() super(CyclocityWebStation, self).update() status_xml = scraper.request(self.url) status = etree.fromstring(status_xml.encode('utf-8')) self.bikes = int(status.findtext('available')) self.free = int(status.findtext('free')) self.extra['open'] = int(status.findtext('open')) == 1 self.extra['last_update'] = status.findtext('updated') self.extra['connected'] = status.findtext('connected') self.extra['slots'] = int(status.findtext('total')) self.extra['ticket'] = int(status.findtext('ticket')) == 1

Compendium ferculorum, albo Zebranie potraw Compendium ferculorum, albo Zebranie potraw (A Collection of Dishes) is a cookbook by Stanisław Czerniecki. First put in print in 1682, it is the earliest cookery book published originally in Polish. Czerniecki wrote it in his capacity as head chef at the court of the house of Lubomirski and dedicated it to Princess Helena Tekla Lubomirska. The book contains more than 300 recipes, divided into three chapters of about 100 recipes each. The chapters are devoted, respectively, to meat, fish and other dishes, and each concludes with a "master chef's secret". Czerniecki's cooking style, as is evident in his book, was typical for the luxuriant Polish Baroque cuisine, which still had a largely medieval outlook, but was gradually succumbing to novel culinary ideas coming from France. It was characterized by generous use of vinegar, sugar and exotic spices, as well as preference for spectacle over thrift. The book was republished several times during the 18th and 19th centuries, sometimes under new titles, and had an important impact on the development of Polish cuisine. It also served as an inspiration for the portrayal of an Old Polish banquet in Pan Tadeusz, the Polish national epic. Context Czerniecki was an ennobled burgher who served three generations of the magnate house of Princes Lubomirski as property manager and head chef. He began his service in ca. 1645, initially under Stanisław Lubomirski until 1649, then under the latter's eldest son, Aleksander Michał Lubomirski, and his grandson, Józef Karol Lubomirski. Although Czerniecki's book was first published in 1682, it must have been completed before Aleksander Michał Lubomirski's death in 1677. The original publication of Compendium ferculorum came three decades after the French cookbook entitled (The French [male] Cook, 1651), by François Pierre de La Varenne, started a culinary revolution that spread across Europe in the second half of the 17th century. In this new wave of French gastronomy, exotic spices were largely replaced with domestic herbs with the aim of highlighting the natural flavours of foods. Polish magnates often hired French chefs at their courts and some also held copies of in their libraries (including King John III Sobieski who was married to the French Marie Casimire d'Arquien). Compendium ferculorum, written in Polish and promoting traditional domestic cuisine, which maintained a largely medieval outlook, may be seen as Czerniecki's response to the onslaught of culinary cosmopolitanism. Content Title and dedication Although the book is written entirely in Polish, it has a bilingual, Latin and Polish, title. Compendium ferculorum and both mean "a collection of dishes", in Latin and Polish, respectively; these are joined by the Polish conjunction albo, "or". The work opens with Czerniecki's dedication to his "most charitable lady and benefactress", Princess née , recalling a famed banquet given to Pope Urban VIII by her father, Prince , during his diplomatic mission to Rome in 1633. Ossoliński's legation was famous for its ostentatious sumptuousness designed to show off the grandeur and prosperity of the Polish-Lithuanian Commonwealth, even to the point of deliberately fitting his mount with loose golden horseshoes, only to lose them while ceremoniously entering the Eternal City. The dedication, in which Czerniecki pointed out he had already served the house of Lubomirski for 32 years, also mentions Lubomirska's husband, , and their son, , wishing the whole family good health and good fortune. The author was well aware that his was the first cookbook in the Polish vernacular, which he made clear in the dedication. "As no one before me has yet wished to present to the world such useful knowledge in our Polish language," he wrote in his opening line, "I have dared, ... despite my ineptitude, to offer the Polish world my compendium ferculorum, or collection of dishes." Polish bibliographer maintained that Compendium ferculorum was preceded by Kuchmistrzostwo (Cooking Mastery), a 16th-century Polish translation of the Czech cookbook (Cookery) written by in 1535. However, no copies of this supposed translation have survived, except for facsimiles of two folios, published in 1891, of uncertain authenticity. Whatever the case, Czerniecki's work is without a doubt the oldest known cookbook published originally in Polish. Inventory The dedication is followed by a detailed inventory of food items, kitchenware, as well as kitchen and waiting staff, necessary for hosting a banquet. The list of food items begins with meats of farm animals and game, including sundry game birds, from snow bunting to great bustard. Different kinds of cereals and pasta are followed by an enumeration of fruits and mushrooms which may be either fresh or dried. The list of vegetables includes, now largely forgotten, cardoon, Jerusalem artichoke and turnip-rooted chervil, or popie jajka (literally, "priest's balls"), as Czerniecki calls it. Under the heading of "spices" come not only saffron, black pepper, ginger, cinnamon, cloves, nutmeg, mace and cumin, all of which were used abundantly in Czerniecki's cookery, but also powdered sugar, rice, "large" and "small" raisins, citrus fruits such as lemons, limes and oranges, and even smoked ham and smoked beef tongue, which were also used as seasonings. Advice Next, the author presents his views and advice regarding the role of kuchmistrz, or "master chef". Czerniecki clearly took pride from his role as a chef, which he understood as incorporating those of an artist and a mentor to younger cooks. According to him, a good chef should be "well-groomed, sober, attentive, loyal and, most of all, supportive to his lord and quick." He should be neat and tidy, with a good head of hair, well-combed, short at the back and sides; he should have clean hands, his fingernails should be trimmed, he should wear a white apron; he should not be quarrelsome, he should be sober, submissive, brisk; he should have a good understanding of flavor, a sound knowledge of ingredients and utensils, together with a willingness to serve everyone. Czerniecki concludes these introductory remarks with an admonition against sprinkling food with bread crumbs and by avowing to focus on "Old Polish dishes" to let the reader experience domestic cuisine before moving on to foreign specialties. Structure The main part of the book follows a well-thought-out structure. It is divided into three chapters, each containing one hundred numbered recipes, followed by an appendix (additament) of ten additional numbered recipes and an extra special recipe called a "master chef's secret" (sekret kuchmistrzowski). Thus, ostensibly, each chapter consists of 111 recipes, giving a total of 333. The first chapter, opening with a recipe for "Polish broth" (rosół polski), is devoted to meat dishes. The second contains recipes for fish dishes, including one for beaver tail, which was considered a fish for the purposes of Catholic dietary law. The final chapter is more diverse and presents recipes for dairy dishes, pies, tarts and cakes, as well as soups (omitted from the chapter's title). The actual number of recipes deviates from that claimed by the author. On the one hand, there may be more than one recipe under a single numbered heading. For example, under number 4 in chapter 1, there are three disparate recipes, one for stewed meat in saffron sauce, one for a thick sauce of sieved vegetables and one for boiled meatballs, as well as a tip to add raisins only to those dishes that are meant to be sweet. Chapter 2 opens with five unnumbered recipes for sauces to be used with fish. On the other hand, some of the numbered headings are not followed by any actual recipe, serving only to reiterate that various dishes may be prepared using the same basic technique. For instance, in chapter 3, the recipe for a puff-pastry apple tart, is followed by eleven numbered headings saying, "pear tart likewise", "woodland strawberry tart likewise", "sour cherry tart likewise", and so on for fig, prune, date, gooseberry, peach, plum, currant and quince tarts. Approach Writing The instructions in Compendium ferculorum are succinct and often vague, lacking such elements of modern culinary recipes as lists of ingredients used, measurements or proportions, and cooking times. The underlying assumption is that they are to be used by a professional chef rather than a person with little cooking experience. The recipe for stewed meat in saffron sauce, cited below, is typical is this regard. All recipes are written in the second-person singular imperative, a grammatical form that would not have been used to address a person of high rank. The instructions were most likely meant to be read aloud by the chef or one of senior cooks to junior members of the kitchen staff, who would carry them out. This style contrasts with the less direct, impersonal way of addressing the reader, characterized by infinitive verb forms, used in 19th-century Polish cookbooks. Cooking Czerniecki's cooking style, as presented in his book, is characterized by aristocratic lavishness, Baroque pageantry and fiery combinations of contrasting flavours. When it comes to spending, the author cautions against both waste and unnecessary thrift. A banquet was meant to overwhelm the guests with lavishness and flaunt the host's wealth and munificence. Abundant use of expensive spices was one way to accomplish this. Black pepper, ginger, saffron, cinnamon, cloves, mace and nutmeg were added to most dishes by kilograms. Czerniecki counted sugar among spices and used it as such; his book contains few recipes for desserts, but sugar is used profusely in recipes for meat, fish and egg dishes. Vinegar was also used in copious amounts. Such fusion of excessively piquant, sweet and sour tastes, which modern Poles would likely find inedible, was typical of Old Polish cuisine, described by Czerniecki as "saffrony and peppery". A dish that succeeded to puzzle or surprise the diners was considered the greatest achievement of culinary sophistication. The "secrets", divulged at the end of each chapter, provide examples of such recipes. One is for a capon in a bottle; the trick was to skin the bird, place the skin inside a bottle, fill it with a mixture of milk and eggs, and put the bottle into boiling water. As the mixture expanded in heat, it produced an illusion of a whole capon fit inside a bottle. Another "secret" is a recipe for an uncut pike with its head fried, its tail baked and its middle boiled; it was made by spit-roasting the fish while basting its head with oil or butter and sprinkling it with flour, and pouring salted vinegar on cloth wrapped around the middle part. The last recipe is for a broth boiled with a string of pearls and a golden coin, which was supposed to cure the sick and those "despairing about their health". Czerniecki's goal was to present what he called "Old Polish dishes" as opposed to foreign recipes. He did, however, reference foreign influences throughout the book, particularly from French cuisine and from what he referred to as "Imperial cuisine", that is, the cooking styles of Bohemia and Hungary, then both part of the Habsburg Empire. He displays an ambivalent attitude to French cookery; on the one hand, he dismisses French , or creamy soups, as alien to Polish culture and criticizes the use of wine in cooking. "Every dish may be cooked without wine", he wrote, "it suffices to season it with vinegar and sweetness." On the other hand, Czerniecki does provide recipes for , dishes seasoned with wine, as well as other French culinary novelties, such as puff pastry (still known today in Poland as "French dough"), arguing that a skilled chef must be able to accommodate foreign visitors with dishes from their own cuisines. In fact, the author included more French recipes than he was ready to admit; even the "secret" recipe for a capon in a bottle was adapted from a similar recipe found in . Legacy Culinary For over a century after its first edition, Compendium ferculorum remained the only printed cookbook in Polish. It was only in 1783 that a new Polish-language cookbook – Kucharz doskonały (The Perfect [male] Cook) by Wojciech Wielądko – was published. Its first edition was little more than an abridged translation of La cuisinière bourgeoise (The Urban [female] Cook) by Menon, a popular French cookbook, first published in 1746. One part of Kucharz doskonały, however, was based directly on Czerniecki's work. Wielądko chose not to translate the glossary of culinary terms included in Menon's book; instead, he appended to the cookbook his own glossary, which explained the culinary terms (and, in some cases, quoted entire recipes) found in Compendium ferculorum. This way, Kucharz doskonały combined French cuisine, which was then storming into fashion in Poland, with Old Polish cookery, exemplified by Czerniecki's book. In the third edition, in 1800, Wielądko included even more time-honoured Polish and German recipes, creating an amalgam of national culinary tradition and new French cuisine, and thus laying the groundwork for modern Polish cookery. Literary In 1834, Compendium ferculorum served as an inspiration to Adam Mickiewicz for his nostalgic description of "the last Old Polish feast" in Pan Tadeusz, a mock heroic poem set in the years 1811–1812, which has come to be revered as the Polish national epic. In his account of the fictional banquet in Book 12, the poet included the names of several dishes found in the oldest Polish cookbook, such as "royal borscht", as well as two of the master chef's secrets: the broth with pearls and a coin, and the three-way fish. To underscore that the feast represents an exotic bygone world of pre-partition Poland, Mickiewicz added to this a list of random dishes, ingredients and additives, whose names he found in Compendium ferculorum and which had already been forgotten in his own time: kontuza (soup of boiled and sieved meat), arkas (sweet milk-based jelly), blemas (blancmange), pomuchla (Atlantic cod), figatele (meatballs), cybeta (civet), piżmo (musk), dragant (tragacanth), pinele (pine nuts) and brunele (prunes). This is followed by a similar enumeration of various kinds of fish, whose names are likewise taken from the inventory at the beginning of Czerniecki's book. A description of a white-apron-clad chef and a reference to Prince Ossoliński's banquet in Rome, both found in Book 11, are also clearly inspired by Czerniecki's characterization of a master chef and his dedication, respectively. What is intriguing about this literary link between the oldest Polish cookbook and the national epic is that Mickiewicz apparently confused the title of Czerniecki's work with that of Wielądko's Kucharz doskonały, both in the poem itself and in the poet's explanatory notes. Whether this is a result of the author's mistake or poetic license, remains a matter of scholarly dispute. According to the poet's friend, Antoni Edward Odyniec, Mickiewicz never parted with an "old and torn book" entitled Doskonały kucharz, which he carried in his personal traveling library. Stanisław Pigoń, a historian of Polish literature, has suggested that it was actually a copy of Czerniecki's book, which happened to be missing its title page, so Mickiewicz was familiar with the contents of Compendium ferculorum, but not with its title, and confounded it with Wielądko's cookbook. Publishing history The original edition of 1682 was the only publication of Czerniecki's cookbook during his lifetime. After his death, however, Compendium ferculorum was republished about 20 times until 1821. Some of the later editions were published under new titles, reflecting a broadening of the target readership. The second edition, published in 1730, was very similar to the first, with only minor revisions. Both editions were printed in blackletter type and in quarto book format; later editions were printed in roman type and in octavo. The third edition, published in Wilno (now Vilnius, Lithuania) in 1744, was the first to be printed outside Kraków. It was also the first edition to be given a new title, Stół obojętny, to jest pański, a oraz i chudopacholski (The Indifferent Table, that is, Both Lordly and Common). The Sandomierz edition of 1784, under the latter title, was expanded with forty new recipes for sauces, gingerbread, vinegar and other condiments. This addition, copied from Compendium medicum, a popular 18th-century Polish medical reference book, differed in both writing and cooking style from Czerniecki's original work, and did not reappear in subsequent editions. Editions from the first quarter of the 19th century, published in Warsaw and Berdyczów (now Berdychiv, Ukraine) use yet another title, Kucharka miejska i wiejska (The Urban and Rural [female] Cook). These changes in title indicate that the cooking style promoted by the Lubomirskis' head chef, originally associated with fine dining at a magnate court, eventually became part of the culinary repertoire of housewives in towns and countryside throughout Poland. Renewed popular interest in Old Polish cuisine has resulted in reprints being made since the turn of the 21st century, including a limited bibliophile edition of 500 numbered leather-bound volumes published in 2002 in Jędrzejów. In 2009, the Wilanów Palace Museum in Warsaw published a critical edition with a broad introduction by food historian Jarosław Dumanowski presenting Czerniecki's life and work. List of editions (2nd ed. 1775, 3rd ed. 1782, 4th ed. 1788) (2nd ed. 1757, ) Enlarged with 40 additional recipes. (2nd ed. 1804, 3rd ed. 1806 in Berdyczów, 4th ed. 1811, 5th ed. 1816, 6th ed. 1821) (2nd ed. 2004) Limited bibliophile edition. (2nd ed. 2010, 3rd ed. 2012) Photo-offset reprint of the 1682 edition. Translations In the late 17th century, Compendium ferculorum was anonymously translated into Russian as (Cookery Book). This translation was never published and is only known from a manuscript held at the Russian State Library in Moscow (another manuscript Russian translation of Czerniecki's work is possibly held at the National Library of Russia in Saint Petersburg). In 2014, an English version of the Wilanów edition was published; the publication of a French translation is also planned. Notes References Sources in . External links Compendium ferculorum, a complete scan of the first edition at Polona.pl, a digital library run by the National Library of Poland Category:1682 books Category:Early Modern cookbooks Category:Polish cuisine

Winning the Culture War is an exposé on America s gradual but steady secular and socialist drift. It serves as a treatise that unashamedly reinforces our country s Judeo-Christian heritage while sounding the alarm at where we as Americans are heading unless Christians and concerned citizens take a firm stand.

Replica-Gun Safety Tips for Youth *If a law-enforcement officer approaches while you’re carrying a replica weapon, immediately stop and follow his or her commands. *Make sure everyone who can see you with this type of gun knows you’re playing a game with a pneumatic gun – Airsoft, BB or pellet. *Obey all laws when carrying or using these weapons; there’s a difference between role-playing with these guns and actually firing them. *Obtain permission from the property owner before playing with replica weapons on his or her property. *Never remove, alter or disguise the weapon’s orange safety tip. *Always follow safety rules for firearms handling. Besides police and sheriffs, some Fairfax fathers and sons also attended last week’s press conference on replica weapons. The sons hadn’t done anything wrong, but they regularly use these weapons to play a popular game called Airsoft. Bill Wilkinson, a P.E. and health teacher at Lanier Middle School, brought his son Brady, 14, a rising freshman at Fairfax High. Wilkinson said his older son Kyle, 16, plays the game, too. "A couple guys on each team work together to eliminate the other people," explained Brady. "You’re using replica weapons that look like real guns like pistols and sniper rifles. We play in the woods or in our backyard." A few months ago, said Wilkinson, "There was incident in the community where some kids had been seen carrying weapons, and citizens called Fairfax City police. The police later gathered the names of local kids known to play Airsoft." Next thing Wilkinson knew, Fairfax police Sgt. Kyle Penman was at his home, interviewing him, his wife and Brady. "We were quite surprised," said Wilkinson. "He explained what was going on, and it was educational for us and brought home the dangers of using Airsoft weapons. We were aware, but it made us more so, and it created a dialogue between my wife and sons and I." Brady said he understood why neighbors might be concerned because "I carry the guns bicycling to a friend’s house or sticking out of my backpack. I’m now more careful carrying them around; I conceal them better." Regarding the press conference, Wilkinson said, "I’m a huge advocate of the police. So to be able to help them out to put this [information] in front of the news was really important to us. And as a teacher, I know education is key." Dad Stan Tomajko came with his son Jordan, 14, also a rising Fairfax High freshman. "I brought my [replica] M14 rifle," said Jordan. "We play Airsoft in our backyards; the weapons fire plastic BBs that sting, but we’re always careful about how we use them." Jordan said Friday’s event taught him that, "If an officer ever sees you with one of these weapons, you should always put it down and listen to what he says. That way, he doesn’t perceive you as a threat." Tomajko came because, he said, "It’s important for me to be educated on what police perceive is the real thing. I wanted my son to learn that, as well."

NEW ORLEANS — The success of the read-option this season has sparked heated debate over whether it’s a passing fad or something that’s here to stay. The Baltimore Ravens are charged with finding ways to combat 49ers quarterback Colin Kaepernick whenever he lines up in the pistol formation and runs the read-option Sunday in the Super Bowl. Ravens middle linebacker Ray Lewis said the 49ers succeed so often using the read-option because defenses aren’t well-prepared. “They’re doing a good job with it,” Lewis said of the 49ers. “At the same time, when you do watch the film, a lot of people who played against them just never communicated at all. “That’s one of the advantages of what we have as a defense. We do a job of communicating very well, whether you have the dive, whether you have the quarterback, how are you going to play this, how are you going to play that?” Lewis said something glaring stands out when he breaks down tape of how other defenses play the read-option. “You can tell that a lot of people playing against the read-option just played as individuals,” Lewis said. “It’s really hard to play that type of package as individuals. You have to play it as a group. “If you (want) to try to slow it down, that’s the only way to slow it down is to really play it as a group and make sure before the snap that everyone is on the same page.” Ravens coach John Harbaugh said he is a big fan of the pistol formation first implemented by Chris Ault, Kaepernick’s college coach at Nevada. “The beauty of it is, and part of the genius of it is, it’s such a simple idea,” Harbaugh said. “It goes back to Nevada and coach Ault out there. You can run your whole offense out of it. You’re not limited to an option type of attack out of it. It’s just a very versatile kind of offense.” Lewis sees plenty of similarities between the way he plays and the way 49ers inside linebacker Patrick Willis plays. He also draws a comparison to the 49ers and some of the great Ravens defenses of the past 15 years or so. “You see the youth of what we were like when we were younger,” Lewis said of the 49ers. “They’re just running around and they’re making a lot of plays. They have a lot of young guys who just love playing the game. “You can tell that they really enjoy playing, not just the game, but playing with each other. And that’s one of the keys to playing great defense is just having that defensive chemistry.” More in News Thirty-six people — musicians, artists, students, lovers and friends — lost their lives on Dec. 2, 2016, in the fire that consumed the Oakland warehouse known as the “Ghost Ship.” Here are their stories. A long-awaited plan to keep the Raiders in Oakland was announced late Friday by city and council officials. It includes a public investment of $350 million, pegged to the value of the Coliseum land and infrastructure improvements.

// Mantid Repository : https://github.com/mantidproject/mantid // // Copyright &copy; 2018 ISIS Rutherford Appleton Laboratory UKRI, // NScD Oak Ridge National Laboratory, European Spallation Source, // Institut Laue - Langevin & CSNS, Institute of High Energy Physics, CAS // SPDX - License - Identifier: GPL - 3.0 + #include "ISISCalibration.h" #include "MantidAPI/MatrixWorkspace.h" #include "MantidAPI/WorkspaceGroup.h" #include "MantidGeometry/Instrument.h" #include "MantidKernel/Logger.h" #include <QDebug> #include <QFileInfo> #include <stdexcept> using namespace Mantid::API; using namespace MantidQt::MantidWidgets; namespace { Mantid::Kernel::Logger g_log("ISISCalibration"); template <typename Map, typename Key, typename Value> Value getValueOr(const Map &map, const Key &key, const Value &defaultValue) { try { return map.at(key); } catch (std::out_of_range &) { return defaultValue; } } } // namespace using namespace Mantid::API; using MantidQt::API::BatchAlgorithmRunner; namespace MantidQt { namespace CustomInterfaces { //---------------------------------------------------------------------------------------------- /** Constructor */ ISISCalibration::ISISCalibration(IndirectDataReduction *idrUI, QWidget *parent) : IndirectDataReductionTab(idrUI, parent), m_lastCalPlotFilename("") { m_uiForm.setupUi(parent); setOutputPlotOptionsPresenter(std::make_unique<IndirectPlotOptionsPresenter>( m_uiForm.ipoPlotOptions, this, PlotWidget::SpectraBin)); m_uiForm.ppCalibration->setCanvasColour(QColor(240, 240, 240)); m_uiForm.ppResolution->setCanvasColour(QColor(240, 240, 240)); m_uiForm.ppCalibration->watchADS(false); m_uiForm.ppResolution->watchADS(false); auto *doubleEditorFactory = new DoubleEditorFactory(); // CAL PROPERTY TREE m_propTrees["CalPropTree"] = new QtTreePropertyBrowser(); m_propTrees["CalPropTree"]->setFactoryForManager(m_dblManager, doubleEditorFactory); m_uiForm.propertiesCalibration->addWidget(m_propTrees["CalPropTree"]); // Cal Property Tree: Peak/Background m_properties["CalPeakMin"] = m_dblManager->addProperty("Peak Min"); m_properties["CalPeakMax"] = m_dblManager->addProperty("Peak Max"); m_properties["CalBackMin"] = m_dblManager->addProperty("Back Min"); m_properties["CalBackMax"] = m_dblManager->addProperty("Back Max"); m_propTrees["CalPropTree"]->addProperty(m_properties["CalPeakMin"]); m_propTrees["CalPropTree"]->addProperty(m_properties["CalPeakMax"]); m_propTrees["CalPropTree"]->addProperty(m_properties["CalBackMin"]); m_propTrees["CalPropTree"]->addProperty(m_properties["CalBackMax"]); // Cal plot range selectors auto calPeak = m_uiForm.ppCalibration->addRangeSelector("CalPeak"); calPeak->setColour(Qt::red); auto calBackground = m_uiForm.ppCalibration->addRangeSelector("CalBackground"); calBackground->setColour(Qt::blue); // blue to be consistent with fit wizard // RES PROPERTY TREE m_propTrees["ResPropTree"] = new QtTreePropertyBrowser(); m_propTrees["ResPropTree"]->setFactoryForManager(m_dblManager, doubleEditorFactory); m_uiForm.loResolutionOptions->addWidget(m_propTrees["ResPropTree"]); // Res Property Tree: Spectra Selection m_properties["ResSpecMin"] = m_dblManager->addProperty("Spectra Min"); m_propTrees["ResPropTree"]->addProperty(m_properties["ResSpecMin"]); m_dblManager->setDecimals(m_properties["ResSpecMin"], 0); m_properties["ResSpecMax"] = m_dblManager->addProperty("Spectra Max"); m_propTrees["ResPropTree"]->addProperty(m_properties["ResSpecMax"]); m_dblManager->setDecimals(m_properties["ResSpecMax"], 0); // Res Property Tree: Background Properties QtProperty *resBG = m_grpManager->addProperty("Background"); m_propTrees["ResPropTree"]->addProperty(resBG); m_properties["ResStart"] = m_dblManager->addProperty("Start"); resBG->addSubProperty(m_properties["ResStart"]); m_properties["ResEnd"] = m_dblManager->addProperty("End"); resBG->addSubProperty(m_properties["ResEnd"]); // Res Property Tree: Rebinning const int NUM_DECIMALS = 3; QtProperty *resRB = m_grpManager->addProperty("Rebinning"); m_propTrees["ResPropTree"]->addProperty(resRB); m_properties["ResELow"] = m_dblManager->addProperty("Low"); m_dblManager->setDecimals(m_properties["ResELow"], NUM_DECIMALS); m_dblManager->setValue(m_properties["ResELow"], -0.2); resRB->addSubProperty(m_properties["ResELow"]); m_properties["ResEWidth"] = m_dblManager->addProperty("Width"); m_dblManager->setDecimals(m_properties["ResEWidth"], NUM_DECIMALS); m_dblManager->setValue(m_properties["ResEWidth"], 0.002); m_dblManager->setMinimum(m_properties["ResEWidth"], 0.001); resRB->addSubProperty(m_properties["ResEWidth"]); m_properties["ResEHigh"] = m_dblManager->addProperty("High"); m_dblManager->setDecimals(m_properties["ResEHigh"], NUM_DECIMALS); m_dblManager->setValue(m_properties["ResEHigh"], 0.2); resRB->addSubProperty(m_properties["ResEHigh"]); // Res plot range selectors // Create ResBackground first so ResPeak is drawn above it auto resBackground = m_uiForm.ppResolution->addRangeSelector("ResBackground"); resBackground->setColour(Qt::blue); auto resPeak = m_uiForm.ppResolution->addRangeSelector("ResPeak"); resPeak->setColour(Qt::red); // SIGNAL/SLOT CONNECTIONS // Update instrument information when a new instrument config is selected connect(this, SIGNAL(newInstrumentConfiguration()), this, SLOT(setDefaultInstDetails())); #if QT_VERSION < QT_VERSION_CHECK(5, 0, 0) connect(resPeak, SIGNAL(rangeChanged(double, double)), resBackground, SLOT(setRange(double, double))); #endif // Update property map when a range selector is moved connect(calPeak, SIGNAL(minValueChanged(double)), this, SLOT(calMinChanged(double))); connect(calPeak, SIGNAL(maxValueChanged(double)), this, SLOT(calMaxChanged(double))); connect(calBackground, SIGNAL(minValueChanged(double)), this, SLOT(calMinChanged(double))); connect(calBackground, SIGNAL(maxValueChanged(double)), this, SLOT(calMaxChanged(double))); connect(resPeak, SIGNAL(minValueChanged(double)), this, SLOT(calMinChanged(double))); connect(resPeak, SIGNAL(maxValueChanged(double)), this, SLOT(calMaxChanged(double))); connect(resBackground, SIGNAL(minValueChanged(double)), this, SLOT(calMinChanged(double))); connect(resBackground, SIGNAL(maxValueChanged(double)), this, SLOT(calMaxChanged(double))); // Update range selector positions when a value in the double manager changes connect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); // Plot miniplots after a file has loaded connect(m_uiForm.leRunNo, SIGNAL(filesFound()), this, SLOT(calPlotRaw())); // Toggle RES file options when user toggles Create RES File checkbox connect(m_uiForm.ckCreateResolution, SIGNAL(toggled(bool)), this, SLOT(resCheck(bool))); // Shows message on run button when user is inputting a run number connect(m_uiForm.leRunNo, SIGNAL(fileTextChanged(const QString &)), this, SLOT(pbRunEditing())); // Shows message on run button when Mantid is finding the file for a given run // number connect(m_uiForm.leRunNo, SIGNAL(findingFiles()), this, SLOT(pbRunFinding())); // Reverts run button back to normal when file finding has finished connect(m_uiForm.leRunNo, SIGNAL(fileFindingFinished()), this, SLOT(pbRunFinished())); // Nudge resCheck to ensure res range selectors are only shown when Create RES // file is checked resCheck(m_uiForm.ckCreateResolution->isChecked()); connect(m_batchAlgoRunner, SIGNAL(batchComplete(bool)), this, SLOT(algorithmComplete(bool))); // Handle running, plotting and saving connect(m_uiForm.pbRun, SIGNAL(clicked()), this, SLOT(runClicked())); connect(m_uiForm.pbSave, SIGNAL(clicked()), this, SLOT(saveClicked())); connect(this, SIGNAL(updateRunButton(bool, std::string const &, QString const &, QString const &)), this, SLOT(updateRunButton(bool, std::string const &, QString const &, QString const &))); } //---------------------------------------------------------------------------------------------- /** Destructor */ ISISCalibration::~ISISCalibration() {} std::pair<double, double> ISISCalibration::peakRange() const { return std::make_pair(m_dblManager->value(m_properties["CalPeakMin"]), m_dblManager->value(m_properties["CalPeakMax"])); } std::pair<double, double> ISISCalibration::backgroundRange() const { return std::make_pair(m_dblManager->value(m_properties["CalBackMin"]), m_dblManager->value(m_properties["CalBackMax"])); } std::pair<double, double> ISISCalibration::resolutionRange() const { return std::make_pair(m_dblManager->value(m_properties["ResStart"]), m_dblManager->value(m_properties["ResEnd"])); } QString ISISCalibration::peakRangeString() const { return m_properties["CalPeakMin"]->valueText() + "," + m_properties["CalPeakMax"]->valueText(); } QString ISISCalibration::backgroundRangeString() const { return m_properties["CalBackMin"]->valueText() + "," + m_properties["CalBackMax"]->valueText(); } QString ISISCalibration::instrumentDetectorRangeString() { return getInstrumentDetail("spectra-min") + "," + getInstrumentDetail("spectra-max"); } QString ISISCalibration::outputWorkspaceName() const { auto name = QFileInfo(m_uiForm.leRunNo->getFirstFilename()).baseName(); if (m_uiForm.leRunNo->getFilenames().size() > 1) name += "_multi"; return name + QString::fromStdString("_") + getAnalyserName() + getReflectionName(); } QString ISISCalibration::resolutionDetectorRangeString() const { return QString::number(m_dblManager->value(m_properties["ResSpecMin"])) + "," + QString::number(m_dblManager->value(m_properties["ResSpecMax"])); } QString ISISCalibration::rebinString() const { return QString::number(m_dblManager->value(m_properties["ResELow"])) + "," + QString::number(m_dblManager->value(m_properties["ResEWidth"])) + "," + QString::number(m_dblManager->value(m_properties["ResEHigh"])); } QString ISISCalibration::backgroundString() const { return QString::number(m_dblManager->value(m_properties["ResStart"])) + "," + QString::number(m_dblManager->value(m_properties["ResEnd"])); } void ISISCalibration::setPeakRange(const double &minimumTof, const double &maximumTof) { auto calibrationPeak = m_uiForm.ppCalibration->getRangeSelector("CalPeak"); setRangeSelector(calibrationPeak, m_properties["CalPeakMin"], m_properties["CalPeakMax"], qMakePair(minimumTof, maximumTof)); } void ISISCalibration::setBackgroundRange(const double &minimumTof, const double &maximumTof) { auto background = m_uiForm.ppCalibration->getRangeSelector("CalBackground"); setRangeSelector(background, m_properties["CalBackMin"], m_properties["CalBackMax"], qMakePair(minimumTof, maximumTof)); } void ISISCalibration::setRangeLimits( MantidWidgets::RangeSelector *rangeSelector, const double &minimum, const double &maximum, const QString &minPropertyName, const QString &maxPropertyName) { setPlotPropertyRange(rangeSelector, m_properties[minPropertyName], m_properties[maxPropertyName], qMakePair(minimum, maximum)); } void ISISCalibration::setPeakRangeLimits(const double &peakMin, const double &peakMax) { auto calibrationPeak = m_uiForm.ppCalibration->getRangeSelector("CalPeak"); setRangeLimits(calibrationPeak, peakMin, peakMax, "CalELow", "CalEHigh"); } void ISISCalibration::setBackgroundRangeLimits(const double &backgroundMin, const double &backgroundMax) { auto background = m_uiForm.ppCalibration->getRangeSelector("CalBackground"); setRangeLimits(background, backgroundMin, backgroundMax, "CalStart", "CalEnd"); } void ISISCalibration::setResolutionSpectraRange(const double &minimum, const double &maximum) { m_dblManager->setValue(m_properties["ResSpecMin"], minimum); m_dblManager->setValue(m_properties["ResSpecMax"], maximum); } void ISISCalibration::setup() {} void ISISCalibration::run() { // Get properties const auto filenames = m_uiForm.leRunNo->getFilenames().join(","); const auto outputWorkspaceNameStem = outputWorkspaceName().toLower(); m_outputCalibrationName = outputWorkspaceNameStem + "_calib"; try { m_batchAlgoRunner->addAlgorithm(calibrationAlgorithm(filenames)); } catch (std::exception const &ex) { g_log.warning(ex.what()); return; } // Initially take the calibration workspace as the result m_pythonExportWsName = m_outputCalibrationName.toStdString(); // Configure the resolution algorithm if (m_uiForm.ckCreateResolution->isChecked()) { m_outputResolutionName = outputWorkspaceNameStem + "_res"; m_batchAlgoRunner->addAlgorithm(resolutionAlgorithm(filenames)); if (m_uiForm.ckSmoothResolution->isChecked()) addRuntimeSmoothing(m_outputResolutionName); // When creating resolution file take the resolution workspace as the result m_pythonExportWsName = m_outputResolutionName.toStdString(); } m_batchAlgoRunner->executeBatchAsync(); } /* * Handle completion of the calibration and resolution algorithms. * * @param error If the algorithms failed. */ void ISISCalibration::algorithmComplete(bool error) { if (!error) { std::vector<std::string> outputWorkspaces{ m_outputCalibrationName.toStdString()}; if (m_uiForm.ckCreateResolution->isChecked() && !m_outputResolutionName.isEmpty()) { outputWorkspaces.emplace_back(m_outputResolutionName.toStdString()); if (m_uiForm.ckSmoothResolution->isChecked()) outputWorkspaces.emplace_back(m_outputResolutionName.toStdString() + "_pre_smooth"); } setOutputPlotOptionsWorkspaces(outputWorkspaces); m_uiForm.pbSave->setEnabled(true); } } bool ISISCalibration::validate() { MantidQt::CustomInterfaces::UserInputValidator uiv; uiv.checkFileFinderWidgetIsValid("Run", m_uiForm.leRunNo); auto rangeOfPeak = peakRange(); auto rangeOfBackground = backgroundRange(); uiv.checkValidRange("Peak Range", rangeOfPeak); uiv.checkValidRange("Back Range", rangeOfBackground); uiv.checkRangesDontOverlap(rangeOfPeak, rangeOfBackground); if (m_uiForm.ckCreateResolution->isChecked()) { uiv.checkValidRange("Background", resolutionRange()); double eLow = m_dblManager->value(m_properties["ResELow"]); double eHigh = m_dblManager->value(m_properties["ResEHigh"]); double eWidth = m_dblManager->value(m_properties["ResEWidth"]); uiv.checkBins(eLow, eWidth, eHigh); } QString error = uiv.generateErrorMessage(); if (error != "") g_log.warning(error.toStdString()); return (error == ""); } /** * Sets default spectra, peak and background ranges. */ void ISISCalibration::setDefaultInstDetails() { try { setDefaultInstDetails(getInstrumentDetails()); } catch (std::exception const &ex) { g_log.warning(ex.what()); showMessageBox(ex.what()); } } void ISISCalibration::setDefaultInstDetails( QMap<QString, QString> const &instrumentDetails) { auto const instrument = getInstrumentDetail(instrumentDetails, "instrument"); auto const spectraMin = getInstrumentDetail(instrumentDetails, "spectra-min").toDouble(); auto const spectraMax = getInstrumentDetail(instrumentDetails, "spectra-max").toDouble(); // Set the search instrument for runs m_uiForm.leRunNo->setInstrumentOverride(instrument); // Set spectra range setResolutionSpectraRange(spectraMin, spectraMax); // Set peak and background ranges const auto ranges = getRangesFromInstrument(); setPeakRange(getValueOr(ranges, "peak-start-tof", 0.0), getValueOr(ranges, "peak-end-tof", 0.0)); setBackgroundRange(getValueOr(ranges, "back-start-tof", 0.0), getValueOr(ranges, "back-end-tof", 0.0)); auto const hasResolution = hasInstrumentDetail(instrumentDetails, "resolution"); m_uiForm.ckCreateResolution->setEnabled(hasResolution); if (!hasResolution) m_uiForm.ckCreateResolution->setChecked(false); } /** * Replots the raw data mini plot and the energy mini plot */ void ISISCalibration::calPlotRaw() { QString filename = m_uiForm.leRunNo->getFirstFilename(); // Don't do anything if the file we would plot has not changed if (filename.isEmpty() || filename == m_lastCalPlotFilename) return; m_lastCalPlotFilename = filename; QFileInfo fi(filename); QString wsname = fi.baseName(); int const specMin = hasInstrumentDetail("spectra-min") ? getInstrumentDetail("spectra-min").toInt() : -1; int const specMax = hasInstrumentDetail("spectra-max") ? getInstrumentDetail("spectra-max").toInt() : -1; if (!loadFile(filename, wsname, specMin, specMax)) { emit showMessageBox("Unable to load file.\nCheck whether your file exists " "and matches the selected instrument in the Energy " "Transfer tab."); return; } const auto input = std::dynamic_pointer_cast<MatrixWorkspace>( AnalysisDataService::Instance().retrieve(wsname.toStdString())); m_uiForm.ppCalibration->clear(); m_uiForm.ppCalibration->addSpectrum("Raw", input, 0); m_uiForm.ppCalibration->resizeX(); const auto &dataX = input->x(0); setPeakRangeLimits(dataX.front(), dataX.back()); setBackgroundRangeLimits(dataX.front(), dataX.back()); setDefaultInstDetails(); m_uiForm.ppCalibration->replot(); // Also replot the energy calPlotEnergy(); } /** * Replots the energy mini plot */ void ISISCalibration::calPlotEnergy() { const auto files = m_uiForm.leRunNo->getFilenames().join(","); auto reductionAlg = energyTransferReductionAlgorithm(files); reductionAlg->execute(); if (!reductionAlg->isExecuted()) { g_log.warning("Could not generate energy preview plot."); return; } WorkspaceGroup_sptr reductionOutputGroup = AnalysisDataService::Instance().retrieveWS<WorkspaceGroup>( "__IndirectCalibration_reduction"); if (reductionOutputGroup->isEmpty()) { g_log.warning("No result workspaces, cannot plot energy preview."); return; } MatrixWorkspace_sptr energyWs = std::dynamic_pointer_cast<MatrixWorkspace>( reductionOutputGroup->getItem(0)); if (!energyWs) { g_log.warning("No result workspaces, cannot plot energy preview."); return; } const auto &dataX = energyWs->x(0); QPair<double, double> range(dataX.front(), dataX.back()); auto resBackground = m_uiForm.ppResolution->getRangeSelector("ResBackground"); setPlotPropertyRange(resBackground, m_properties["ResStart"], m_properties["ResEnd"], range); m_uiForm.ppResolution->clear(); m_uiForm.ppResolution->addSpectrum("Energy", energyWs, 0); m_uiForm.ppResolution->resizeX(); calSetDefaultResolution(energyWs); m_uiForm.ppResolution->replot(); } /** * Set default background and rebinning properties for a given instrument * and analyser * * @param ws :: Mantid workspace containing the loaded instrument */ void ISISCalibration::calSetDefaultResolution( const MatrixWorkspace_const_sptr &ws) { auto inst = ws->getInstrument(); auto analyser = inst->getStringParameter("analyser"); if (analyser.size() > 0) { auto comp = inst->getComponentByName(analyser[0]); if (!comp) return; auto params = comp->getNumberParameter("resolution", true); // Set the default instrument resolution if (!params.empty()) { double res = params[0]; const auto energyRange = getXRangeFromWorkspace(ws); // Set default rebinning bounds QPair<double, double> peakERange(-res * 10, res * 10); auto resPeak = m_uiForm.ppResolution->getRangeSelector("ResPeak"); setPlotPropertyRange(resPeak, m_properties["ResELow"], m_properties["ResEHigh"], energyRange); setRangeSelector(resPeak, m_properties["ResELow"], m_properties["ResEHigh"], peakERange); // Set default background bounds QPair<double, double> backgroundERange(-res * 9, -res * 8); auto resBackground = m_uiForm.ppResolution->getRangeSelector("ResBackground"); setRangeSelector(resBackground, m_properties["ResStart"], m_properties["ResEnd"], backgroundERange); } } } /** * Handles a range selector having it's minimum value changed. * Updates property in property map. * * @param val :: New minimum value */ void ISISCalibration::calMinChanged(double val) { auto calPeak = m_uiForm.ppCalibration->getRangeSelector("CalPeak"); auto calBackground = m_uiForm.ppCalibration->getRangeSelector("CalBackground"); auto resPeak = m_uiForm.ppResolution->getRangeSelector("ResPeak"); auto resBackground = m_uiForm.ppResolution->getRangeSelector("ResBackground"); MantidWidgets::RangeSelector *from = qobject_cast<MantidWidgets::RangeSelector *>(sender()); disconnect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); if (from == calPeak) { m_dblManager->setValue(m_properties["CalPeakMin"], val); } else if (from == calBackground) { m_dblManager->setValue(m_properties["CalBackMin"], val); } else if (from == resPeak) { m_dblManager->setValue(m_properties["ResELow"], val); } else if (from == resBackground) { m_dblManager->setValue(m_properties["ResStart"], val); } connect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); } /** * Handles a range selector having it's maximum value changed. * Updates property in property map. * * @param val :: New maximum value */ void ISISCalibration::calMaxChanged(double val) { auto calPeak = m_uiForm.ppCalibration->getRangeSelector("CalPeak"); auto calBackground = m_uiForm.ppCalibration->getRangeSelector("CalBackground"); auto resPeak = m_uiForm.ppResolution->getRangeSelector("ResPeak"); auto resBackground = m_uiForm.ppResolution->getRangeSelector("ResBackground"); MantidWidgets::RangeSelector *from = qobject_cast<MantidWidgets::RangeSelector *>(sender()); disconnect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); if (from == calPeak) { m_dblManager->setValue(m_properties["CalPeakMax"], val); } else if (from == calBackground) { m_dblManager->setValue(m_properties["CalBackMax"], val); } else if (from == resPeak) { m_dblManager->setValue(m_properties["ResEHigh"], val); } else if (from == resBackground) { m_dblManager->setValue(m_properties["ResEnd"], val); } connect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); } /** * Update a range selector given a QtProperty and new value * * @param prop :: The property to update * @param val :: New value for property */ void ISISCalibration::calUpdateRS(QtProperty *prop, double val) { auto calPeak = m_uiForm.ppCalibration->getRangeSelector("CalPeak"); auto calBackground = m_uiForm.ppCalibration->getRangeSelector("CalBackground"); auto resPeak = m_uiForm.ppResolution->getRangeSelector("ResPeak"); auto resBackground = m_uiForm.ppResolution->getRangeSelector("ResBackground"); disconnect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); if (prop == m_properties["CalPeakMin"]) { setRangeSelectorMin(m_properties["CalPeakMin"], m_properties["CalPeakMax"], calPeak, val); } else if (prop == m_properties["CalPeakMax"]) { setRangeSelectorMax(m_properties["CalPeakMin"], m_properties["CalPeakMax"], calPeak, val); } else if (prop == m_properties["CalBackMin"]) { setRangeSelectorMin(m_properties["CalPeakMin"], m_properties["CalBackMax"], calBackground, val); } else if (prop == m_properties["CalBackMax"]) { setRangeSelectorMax(m_properties["CalPeakMin"], m_properties["CalBackMax"], calBackground, val); } else if (prop == m_properties["ResStart"]) { setRangeSelectorMin(m_properties["ResStart"], m_properties["ResEnd"], resBackground, val); } else if (prop == m_properties["ResEnd"]) { setRangeSelectorMax(m_properties["ResStart"], m_properties["ResEnd"], resBackground, val); } else if (prop == m_properties["ResELow"]) { setRangeSelectorMin(m_properties["ResELow"], m_properties["ResEHigh"], resPeak, val); } else if (prop == m_properties["ResEHigh"]) { setRangeSelectorMax(m_properties["ResELow"], m_properties["ResEHigh"], resPeak, val); } connect(m_dblManager, SIGNAL(valueChanged(QtProperty *, double)), this, SLOT(calUpdateRS(QtProperty *, double))); } /** * This function enables/disables the display of the options involved in *creating the RES file. * * @param state :: whether checkbox is checked or unchecked */ void ISISCalibration::resCheck(bool state) { m_uiForm.ppResolution->getRangeSelector("ResPeak")->setVisible(state); m_uiForm.ppResolution->getRangeSelector("ResBackground")->setVisible(state); // Toggle scale and smooth options m_uiForm.ckResolutionScale->setEnabled(state); m_uiForm.ckSmoothResolution->setEnabled(state); } /** * Called when a user starts to type / edit the runs to load. */ void ISISCalibration::pbRunEditing() { updateRunButton(false, "unchanged", "Editing...", "Run numbers are currently being edited."); } /** * Called when the FileFinder starts finding the files. */ void ISISCalibration::pbRunFinding() { updateRunButton(false, "unchanged", "Finding files...", "Searching for data files for the run numbers entered..."); m_uiForm.leRunNo->setEnabled(false); } /** * Called when the FileFinder has finished finding the files. */ void ISISCalibration::pbRunFinished() { if (!m_uiForm.leRunNo->isValid()) updateRunButton( false, "unchanged", "Invalid Run(s)", "Cannot find data files for some of the run numbers entered."); else updateRunButton(); m_uiForm.leRunNo->setEnabled(true); } /** * Handle saving of workspace */ void ISISCalibration::saveClicked() { checkADSForPlotSaveWorkspace(m_outputCalibrationName.toStdString(), false); addSaveWorkspaceToQueue(m_outputCalibrationName); if (m_uiForm.ckCreateResolution->isChecked()) { checkADSForPlotSaveWorkspace(m_outputResolutionName.toStdString(), false); addSaveWorkspaceToQueue(m_outputResolutionName); } m_batchAlgoRunner->executeBatchAsync(); } /** * Handle when Run is clicked */ void ISISCalibration::runClicked() { runTab(); } void ISISCalibration::addRuntimeSmoothing(const QString &workspaceName) { auto smoothAlg = AlgorithmManager::Instance().create("WienerSmooth"); smoothAlg->initialize(); smoothAlg->setProperty("OutputWorkspace", workspaceName.toStdString()); BatchAlgorithmRunner::AlgorithmRuntimeProps smoothAlgInputProps; smoothAlgInputProps["InputWorkspace"] = workspaceName.toStdString() + "_pre_smooth"; m_batchAlgoRunner->addAlgorithm(smoothAlg, smoothAlgInputProps); } IAlgorithm_sptr ISISCalibration::calibrationAlgorithm(const QString &inputFiles) { auto calibrationAlg = AlgorithmManager::Instance().create("IndirectCalibration"); calibrationAlg->initialize(); calibrationAlg->setProperty("InputFiles", inputFiles.toStdString()); calibrationAlg->setProperty("OutputWorkspace", m_outputCalibrationName.toStdString()); calibrationAlg->setProperty("DetectorRange", instrumentDetectorRangeString().toStdString()); calibrationAlg->setProperty("PeakRange", peakRangeString().toStdString()); calibrationAlg->setProperty("BackgroundRange", backgroundRangeString().toStdString()); calibrationAlg->setProperty("LoadLogFiles", m_uiForm.ckLoadLogFiles->isChecked()); if (m_uiForm.ckScale->isChecked()) calibrationAlg->setProperty("ScaleFactor", m_uiForm.spScale->value()); return calibrationAlg; } IAlgorithm_sptr ISISCalibration::resolutionAlgorithm(const QString &inputFiles) const { auto resAlg = AlgorithmManager::Instance().create("IndirectResolution", -1); resAlg->initialize(); resAlg->setProperty("InputFiles", inputFiles.toStdString()); resAlg->setProperty("Instrument", getInstrumentName().toStdString()); resAlg->setProperty("Analyser", getAnalyserName().toStdString()); resAlg->setProperty("Reflection", getReflectionName().toStdString()); resAlg->setProperty("RebinParam", rebinString().toStdString()); resAlg->setProperty("DetectorRange", resolutionDetectorRangeString().toStdString()); resAlg->setProperty("BackgroundRange", backgroundString().toStdString()); resAlg->setProperty("LoadLogFiles", m_uiForm.ckLoadLogFiles->isChecked()); if (m_uiForm.ckResolutionScale->isChecked()) resAlg->setProperty("ScaleFactor", m_uiForm.spScale->value()); if (m_uiForm.ckSmoothResolution->isChecked()) resAlg->setProperty("OutputWorkspace", m_outputResolutionName.toStdString() + "_pre_smooth"); else resAlg->setProperty("OutputWorkspace", m_outputResolutionName.toStdString()); return resAlg; } IAlgorithm_sptr ISISCalibration::energyTransferReductionAlgorithm( const QString &inputFiles) const { auto reductionAlg = AlgorithmManager::Instance().create("ISISIndirectEnergyTransferWrapper"); reductionAlg->initialize(); reductionAlg->setProperty("Instrument", getInstrumentName().toStdString()); reductionAlg->setProperty("Analyser", getAnalyserName().toStdString()); reductionAlg->setProperty("Reflection", getReflectionName().toStdString()); reductionAlg->setProperty("InputFiles", inputFiles.toStdString()); reductionAlg->setProperty("SumFiles", m_uiForm.ckSumFiles->isChecked()); reductionAlg->setProperty("OutputWorkspace", "__IndirectCalibration_reduction"); reductionAlg->setProperty("SpectraRange", resolutionDetectorRangeString().toStdString()); reductionAlg->setProperty("LoadLogFiles", m_uiForm.ckLoadLogFiles->isChecked()); return reductionAlg; } void ISISCalibration::setRunEnabled(bool enabled) { m_uiForm.pbRun->setEnabled(enabled); } void ISISCalibration::setSaveEnabled(bool enabled) { m_uiForm.pbSave->setEnabled(enabled); } void ISISCalibration::updateRunButton(bool enabled, std::string const &enableOutputButtons, QString const &message, QString const &tooltip) { setRunEnabled(enabled); m_uiForm.pbRun->setText(message); m_uiForm.pbRun->setToolTip(tooltip); if (enableOutputButtons != "unchanged") setSaveEnabled(enableOutputButtons == "enable"); } } // namespace CustomInterfaces } // namespace MantidQt

Opioid receptor blockade in rat nucleus tractus solitarius alters amygdala dynorphin gene expression. It has been suggested that an opioidergic feeding pathway exists between the nucleus of the solitary tract (NTS) and the central nucleus of the amygdala. We studied the following three groups of rats: 1) artificial cerebrospinal fluid (CSF) infused in the NTS, 2) naltrexone (100 microg/day) infused for 13 days in the NTS, and 3) artificial CSF infused in the NTS of rats pair fed to the naltrexone-infused group. Naltrexone administration resulted in a decrease in body weight and food intake. Also, naltrexone infusion increased dynorphin, but not enkephalin, gene expression in the amygdala, independent of the naltrexone-induced reduction in food intake. Gene expression of neuropeptide Y in the arcuate nucleus and neuropeptide Y peptide levels in the paraventricular nucleus did not change because of naltrexone infusion. However, naltrexone induced an increase in serum leptin compared with pair-fed controls. Thus chronic administration of naltrexone in the NTS increased dynorphin gene expression in the amygdala, further supporting an opioidergic feeding pathway between these two brain sites.

Q: Should I always mutate the offspring in steady-state selection? Having read the pseudocode for implementing steady-state selection in Genetic Algorithms in Essentials of Metaheuristics and this site, should I always mutate the children or I should subject it to a mutation probability, say 50%? I was a bit worried that population will not converge if I always do mutation at each generation, because the chromosomes are represented by real number values from 0 to 1. A: The behaviour of a Genetic Algorithm is very sensible with respect to the weight values that you heuristically choose. The probability values assigned to each action really depend on the particular genetic scheme that you are applying, the way in which data is represented and affected by mutation and crossover, the initial population and the problem itself. So, in order to heuristically choose the best weight values for your problem, I advice you to keep track of the following data: the evolution of the Entropic Diversity Index (ref: Wikipedia) of your population as time goes by the evolution of the best fitness function value within your population as time goes by the evolution of the average fitness function value within your population as time goes by the rate of new individuals that appear with the same genetic code as existing (or past) individuals Here I attach a plot which shows the evolution of these parameters in one of the earliest executions (I still had to balance it out) of AGER, a custom-made genetic algorithm that is capable of automatically finding the best approximation of the ideal design of an embedded microprocessor based on the simplescalar architecture for executing a given application. (click to enlarge picture) Ideally, you want to set your weight values so that the initial slope is as low as possible, but not up to a point that average fitness function value doesn't improve along several generations (it should be allowed to temporarily worsen, though). The diversity should be kept relatively high as long as possible, while the average value of the fitness function should be grow very slowly (unlike in the picture). The best fitness value, instead, should verified to be a monotonically increasing function. Having said this, a rate of mutation equal to 50% seems to be very high.

Feds charge couple in fraud case May 26, 2007|Staff report Federal officials announced charges Friday against a Miami Lakes couple accused of submitting about $80 million in false Medicare claims for prosthetic limbs and other medical supplies that were never ordered by any doctor or delivered to any patient. Mabel and Abner Diaz ran All-Med Billing Corp., a company that submitted Medicare claims on behalf of third parties. According to the 46-count indictment, the couple received about $56 million as a result of fraudulent billings between 1998 and 2004. Mabel Diaz and an employee, Suleidy Cano, are accused of forging prescriptions for some of the medical equipment. The government is seeking forfeiture of the couple's home, bank accounts and other property, including $32,292 and jewelry seized from their home in 2004.

AZTEC — Construction of a pedestrian bridge to provide access to the Aztec Ruins is set to begin early next month. Brush clearing, trail marking and tree removal began a week ago to make way for the oversized, heavy equipment needed to erect the 400-foot, steel and concrete bridge. Weather permitting, the bridge is expected to be completed by the end of June when work to lay the trail beds will begin. "We're waiting on the drill rig from Salt Lake City to arrive around the first of April to drill holes for the piers for the bridge," said Gary Huffman, project manager for RMCI, Inc., the Albuquerque contractor in charge of the bridge construction. "We'll drill 40 feet or so down to the bedrock to anchor the piers and pour the cement in and add rebar to stabilize them." The North Animas Pedestrian Bridge will connect to trails leading from North Main Avenue in downtown Aztec and from the Aztec Ruins National Monument on either side of the river just upstream of the Hampton Arroyo. The 110,000-pound bridge will be shipped from Minnesota in May and assembled on site, Huffman said. Halfway across the river will be two lookout platforms, so pedestrians can stop and take in the view, Huffman said. "The webbing will be rust-colored steel, which will blend in with the natural colors in this area," Huffman said. "It's going to look really nice and not distract from the natural beauty of this place." The bridge and trails projects are part of Aztec's ongoing efforts to promote the city's parks, trails and cultural resources, said Ed Kotyk, projects manager for the city. The bridge and trails are funded through a $424,828 grant provided through the Federal Transit Administration's Paul S. Sarbanes Transit in Parks program. But the location of the bridge in a flood zone and near unexcavated, underground archaeological sites on the north side of the river has meant far more scrutiny and care than most major urban construction projects. Gary Huffman, RMCI, Inc. project manager, discusses the planned Aztec pedestrian bridge on Wednesday at the construction site off of North Main Avenue in Aztec. (Megan Farmer / The Daily Times) "We're here to enhance the beauty of the area, not tear it up. We have got to keep the cultural and natural resources preserved for our kids' kids, future generations," Huffman said. "Construction has changed a lot with technology and greater understanding. You used to come in, bring out a bunch of (bulldozers) and get it done. Nowadays, we get to be more sensitive to the area (in which) we're in." The bridge was originally to be a far simpler. But concerns raised by Larry Turk, parks superintendent for Aztec Ruins and Chaco Culture National Historical Park, last year called for a longer bridge to adjust for the archeological resources throughout the surface and subsurface in the riparian area. The site of the future Aztec pedestrian bridge is shown on Wednesday off of North Main Avenue in Aztec. (Megan Farmer /The Daily Times) "We have such a sensitive site on the Ruins, or western, side of the river," Huffman said. "We will be staying out of the river area to protect the ecosystem. The project has required a lot of coordination between all parties involved — us, the engineer, the city, the park. It's gone really well." Part of the care came after a biological assessment listed a variety of plants and wildlife that were to be accounted for and protected before any construction could begin. "We cleared about 16 or so cottonwood trees that were in the way of the trail (path)," said Gabe Pena, superintendent for RMCI. "We had to have that work completed before the March 15 deadline, when birds like the (Southwestern) willow flycatcher start coming in (the area) to nest." The breeding season for migratory birds like the flycatcher, an endangered species, generally runs between early March and early September, according to a 2012 biological assessment completed by HDR Engineering, Inc. "The project's still on schedule," Pena said. "Hopefully, they'll start drilling April 1st." James Fenton covers Aztec and Bloomfield for The Daily Times. He can be reached at 505-564-4621. RMCI Inc. Project Manager Gary Huffman points to a blueprint of a planned Aztec pedestrian bridge on Wednesday, at the construction site off of North Main Avenue in Aztec. (Megan Farmer/The Daily Times)

Q: How to return both custom HTTP status code and content? I have an WebApi Controller written in ASP.NET Core and would like to return a custom HTTP status code along with custom content. I am aware of: return new HttpStatusCode(myCode) and return Content(myContent) and I am looking for something along the lines of: return Content(myCode, myContent) or some in built mechanism that already does that. So far I have found this solution: var contentResult = new Content(myContent); contentResult.StatusCode = myCode; return contentResult; is another recommended way of achieving this? A: You can use ContentResult: return new ContentResult() { Content = myContent, StatusCode = myCode }; A: You need to use HttpResponseMessage Below is a sample code // GetEmployee action public HttpResponseMessage GetEmployee(int id) { Employee emp = EmployeeContext.Employees.Where(e => e.Id == id).FirstOrDefault(); if (emp != null) { return Request.CreateResponse<Employee>(HttpStatusCode.OK, emp); } else { return Request.CreateErrorResponse(HttpStatusCode.NotFound, " Employee Not Found"); } } More info here

John Gibson (architect) ''For others with the same name, see John Gibson (disambiguation) John Gibson (2 June 1817 – 23 December 1892) was an English architect born at Castle Bromwich, Warwickshire. Life Gibson was an assistant to Sir Charles Barry and assisted him in the drawings of the Houses of Parliament. Gibson was a prominent bank architect at a time when joint-stock banking was an innovation. His 1847 National Bank of Scotland branch in Glasgow led to perhaps his best-known work, the former National Provincial Bank in Bishopsgate, London, designed in 1862. It was listed Grade I in 1950 and is now known as Gibson Hall. Gibson also designed Todmorden Town Hall which opened in 1875. He also designed Dobroyd Castle near Todmorden and Todmorden Unitarian Church. Gibson is responsible for several churches in and around North Wales, but perhaps his most notable church is St Margaret's in Bodelwyddan, Denbighshire, more popularly known as the Marble Church, Bodelwyddan, consecrated in 1860. The church is a prominent landmark in the lower Vale of Clwyd and is visible for many miles. It lies just off the A55 trunk road. In 1890 Gibson was awarded the Royal Gold Medal for services to architecture. Gibson died of pneumonia on 23 December 1892, at his residence, 13 Great Queen Street, Westminster, and was buried in Kensal Green cemetery on 28 December. References Category:1817 births Category:1892 deaths Category:19th-century English architects Category:Recipients of the Royal Gold Medal Category:Architects from Warwickshire Category:Burials at Kensal Green Cemetery Category:People from Castle Bromwich

Q: Check whether element from variable has a refference in current DOM Assume I have an element in a variable: var element = document.getElementsByTagName("div")[0] // here can be any kind of getting element, e. g. React ref, Chrome's devtools $0, etc. At some point of time my markup is changing (like in SPA), and element from variable has been removed from DOM, but it still available in the element with all properties, such as parentElement, etc. The question is: how to check, if my DOM element from element is present in DOM? I tried to check the element.getBoundingClientRect(), and yes, there are some differences: element that removed from DOM has all the zeroes in his bounding rect. But there is one thing: element with display: none also has all the zeroes in its bounding rect, despite of it is still presents in the DOM (physically, lets say). This is not acceptable in my case, because I need to differ hidden element from removed element. A: You can use contains for this purpose function contains() { const result = document.body.contains(element); console.log(result); } const element = document.getElementById('app'); contains(); element.classList.add('hide'); contains(); element.parentNode.removeChild(element); contains(); .hide { display: none; } <div id="app">App</div>

Forza Motorsport 6 (Official Game Trailer) Named 'Best Racing Game' and 'Best Simulation Game' at Gamescom 2015, Forza Motorsport 6 is the biggest Forza ever. The Forza Motorsport 6 Launch Trailer features a selection of the 460 Forzavista cars available in the full version, all with working cockpits and full damage, and highlights from the game’s 70 hour career mode, including wet weather and night racing, all at a stunning 1080p resolution and 60 frames per second.

Morphology, molecules, and the phylogenetics of cetaceans. Recent phylogenetic analyses of cetacean relationships based on DNA sequence data have challenged the traditional view that baleen whales (Mysticeti) and toothed whales (Odontoceti) are each monophyletic, arguing instead that baleen whales are the sister group of the odontocete family Physeteridae (sperm whales). We reexamined this issue in light of a morphological data set composed of 207 characters and molecular data sets of published 12S, 16S, and cytochrome b mitochondrial DNA sequences. We reach four primary conclusions: (1) Our morphological data set strongly supports the traditional view of odontocete monophyly; (2) the unrooted molecular and morphological trees are very similar, and most of the conflict results from alternative rooting positions; (3) the rooting position of the molecular tree is sensitive to choice of artiodactyls outgroup taxa and the treatment of two small but ambiguously aligned regions of the 12S and 16S sequences, whereas the morphological root is strongly supported; and (4) combined analyses of the morphological and molecular data provide a well-supported phylogenetic estimate consistent with that based on the morphological data alone (and the traditional view of toothed-whale monophyly) but with increased bootstrap support at nearly every node of the tree.

Willie Almond William Almond, known as Willie Almond (born 5 April 1868) was an English footballer who played in The Football League for Accrington, Blackburn Rovers and Northwich Victoria. Blackburn Rovers had a highly successful inaugural league season in 1888-1889. They finished 4th in the League and reached the Semi-Final of the FA Cup. Willie Almond made his League and Club debut, playing at centre-half, on 15 September 1888 at Leamington Road, home of Blackburn Rovers in a match against Accrington. The match was drawn 5-5. When he made his League debut Willie Almond was 20 years and 163 days old; that made him, on the second weekend of League football, Blackburn Rovers' youngest player. Willie Almond scored his debut League goal on 10 November 1888 at Leamington Road, Blackburn against Everton FC. Blackburn Rovers won the match 3-0. Willie Almond missed only one of the 21 League matches played by Blackburn Rovers in season 1888-1889 and scored one League goal. As a centre-half he played in a defence-line that kept three clean sheets and kept the opposition to one-in-a-match on three separate occasions. In season 1888-89 Almond played one match as a wing-half and played in a midfield that achieved a big three-goal win. References Category:1868 births Category:English footballers Category:Blackburn Rovers F.C. players Category:Accrington F.C. players Category:Northwich Victoria F.C. players Category:English Football League players Category:Year of death missing Category:Association football midfielders

An outcome study of Missouri's CSTAR alcohol and drug abuse programs. The Comprehensive Substance Abuse Treatment and Rehabilitation (CSTAR) program is described, and a study of its services is presented. The CSTAR program is a community program with wrap-around services and intensive case management. Eleven domains typically affected by substance abuse were measured, plus satisfaction with treatment services. A retrospective study of 280 clients at 10 facilities was done, and results analyzed separately by General Programs. Women with Children programs, and Adolescent programs. A small sample of clients who were early in their treatment was re-interviewed 90 days later. Data were also examined according to length of stay in the program. Results were consistently positive, and increased with length of time in the program.

What remains of non-syndromic bicoronal synostosis? More and more genetic syndromes are associated with bicoronal synostosis (BCS), making non-syndromic BCS (NSBCS) a shrinking entity. However, the numerical importance and clinical impact of syndromic BCS (SBCS) versus NSBCS have not been much studied. We retrospectively reviewed our experience with BCS over the last four decades in order to compare prevalence trends in SBCS and NSBCS. 195 patients were treated for BCS during the period 1978-2017: 104 (53.3%) were syndromic, 24 (12.3%) showed clinical and/or familial features suggesting a syndrome, although without final diagnostic confirmation, and 7 (3.5%) had associated extra-cranial malformations suggesting a syndromic context without identified genetic mutation; the remaining 61 (31.3%) were purely NSBCS. Surgery was required earlier in SBCS (21.7months, 95%CI 18.4-25.1) than in NSBCS (29.5months 95%CI 26.4-32.7). Prevalence of hydrocephalus and tonsillar herniation was significantly lower in NSBCS, and mortality concerned only SBCS. Prevalence of NSBC decreased significantly over the study period, likely because of more accurate testing, and decreased slightly over the last decade, possibly because of prenatal testing and abortion. NSBCS is now much less common than SBCS, and has a less aggressive clinical course, with lower rates of hydrocephalus, tonsillar herniation and mortality. This subgroup also deserves attention because it is likely that new discoveries are still to be made.

Direct observation of the ionization step in solvolysis reactions: electrophilicity versus electrofugality of carbocations. Rates and equilibria of the reactions of highly stabilized amino-substituted benzhydrylium ions (Ar2CH+) with carboxylate ions have been determined photometrically in acetone and acetonitrile solutions. Treatment of covalent benzhydryl carboxylates (Ar2CH-O2CR) with aqueous acetone or acetonitrile leads to the regeneration of the colored amino-substituted benzhydrylium ions Ar2CH+, which do not undergo subsequent reactions with the solvent. One can, therefore, directly measure the first step of S(N)1 reactions. The electrofugality order, i.e., the relative ionization rates of benzhydryl esters Ar2CH-O2CR with the same anionic leaving group, does not correlate with the corresponding electrophilicity order, i.e., the relative reactivities of the corresponding benzhydrylium ions Ar2CH+ toward a common nucleophile. Thus, benzhydrylium ions which are produced with equal rates by ionization of the corresponding covalent esters may differ by more than 2 orders of magnitude in their reactivities toward nucleophiles, e.g., carboxylate ions. Variable intrinsic barriers account for the breakdown of the rate-equilibrium relationships. Complete free-energy profiles for the ionization of benzhydryl carboxylates Ar2CH-O2CR are constructed, which demonstrate that the transition states of these ionizations are not carbocation-like. As a consequence, variation of the solvent-ionizing power Y has only a small effect on the ionization rate constant (m = 0.35 to 0.55) indicating that small values of m in the Winstein-Grunwald equation do not necessarily imply an S(N)2 type mechanism.

Q: printing symbols onto an image? I want to ask that how can i add symbols like 'x' or 'o' onto an image permanently? I know how to write it in a function figure; imshow(I) hold on plot(CentroidTermX,CentroidTermY,'ro','linewidth',2) plot(CentroidBifX,CentroidBifY,'go','linewidth',2) but i want to use the resultant image in another file and also in displaying it in a gui but when i use the I as an output argument function I = detection(X,Y) it gives the original image without the 'ro' and 'go'. What could be the way to do it? Help will be appreciated. A: There are functions for that in the Computer Vision System Toolbox: insertMarker insertShape insertText insertObjectAnnotation

Photocatalytic hydrogen generation using a nanocomposite of multi-walled carbon nanotubes and TiO2 nanoparticles under visible light irradiation. Nanocomposite catalysts (MWNT-TiO(2)) were prepared hydrothermally from multi-walled carbon nanotubes (MWNTs) and titanium sulfate as the titanium source, and then systematically analyzed using electron microscopy, Raman, FT-IR, and UV-vis spectroscopy. Pt-loaded nanocomposites, pristine TiO(2) and MWNTs were examined for their photocatalytic activity on splitting water with triethanolamine as an electron donor. Under visible light irradiation (lambda>420 nm), hydrogen was successfully produced over the Pt/MWNT-TiO(2), while no capacity to split water showed on the Pt-loaded pristine TiO(2) and MWNTs. Under full spectral irradiation of a Xe-lamp, a hydrogen generation rate of up to 8 mmol g(-1) h(-1) or more was achieved. The significant photocatalytic activity of the nanocomposites was attributed to the synergetic effect of the intrinsic properties of its components such as an excellent light absorption and charge separation on the interfaces between the modified MWNTs and TiO(2), resulting from direct growth of TiO(2) nanoparticles on the surface of the MWNTs during the hydrothermal process.

PhKv a toxin isolated from the spider venom induces antinociception by inhibition of cholinesterase activating cholinergic system. Cholinergic agents cause antinociception by mimicking the release of acetylcholine (ACh) from spinal cholinergic nerves. PhKv is a peptide isolated from the venom of the armed spider Phoneutria nigriventer. It has an antiarrythmogenic activity that involves the enhanced release of acetylcholine. The aim of this study was to investigate whether PhKv had an antinociceptive action in mice. Male albino Swiss mice (25-35g) were used in this study. The PhKv toxin was purified from a PhTx3 fraction of the Phoneutria nigriventer spider's venom. Because of its peptide nature, PhKv is not orally available and it was delivered directly into the central nervous system by an intrathecal (i.t.) route. PhKV on the thermal and mechanical sensitivity was evaluated using plantar test apparatus and the up-and-down method. The analgesic effects of PhKv were studied in neuropathic pain (CCI) and in the peripheral capsicin test. In order to test whether PhKv interfered with the cholinergic system, the mice were pre-treated with atropine (5mg/kg, i.p.) or mecamylamine (0.001mg/kg, i.p.) and the PhKv toxin (30pmol/site i.t.) or neostigmine (100pmol/site) were applied 15min before the intraplantar capsaicin (1nmol/paw) administrations. To investigate PhKv action on the AChE activities, was performed in vitro and ex vivo assay for AChE. For the in vitro experiments, mice spinal cord supernatants of tissue homogenates (1mg/ml) were used as source of AChE activity. The AChE assay was monitored at 37°C for 10min in a FlexStation 3 Multi-Mode Microplate Reader (Molecular Devices) at 405nm. PhKv (30 and 100pmol/site, i.t.) had no effect on the thermal or mechanical sensitivity thresholds. However, in a chronic constriction injury model of pain, PhKv (10pmol/site, i.t.) caused a robust reduction in mechanical withdrawal with an antinociceptive effect that lasted 4h. A pretreatment in mice with PhKv (30pmol/site, i.t.) or neostigmine (100pmol/site, i.t.) 15min before an intraplantar injection of capsaicin (1nmol/paw) caused a maximal antinociceptive effect of 69.5±4.9% and 85±2.5%, respectively. A pretreatment in mice with atropine; 5mg/kg, i.p. or mecamylamine 0.001mg/kg, i.p. inhibited a neostigimine and PhKv-induced antinociception, suggesting a cholinergic mechanism. Spinal acetylcholinesterase was inhibited by PhKv with ED50 of 7.6 (4.6-12.6pmol/site, i.t.). PhKv also inhibited the in vitro AChE activity of spinal cord homogenates with an EC50 of 20.8 (11.6-37.3nM), shifting the Km value from 0.06mM to 18.5mM, characterizing a competitive inhibition of AChE activity by PhKv. Our findings provide, to our knowledge, the first evidence that PhKv caused inhibition of AChE, it increased the ACh content at the neuronal synapses, leading to an activation of the cholinergic system and an antinociceptive response. Studies regarding the nociceptive mechanisms and the identification of potential targets for the treatment of pain have become top priorities. PhKv, by its action of stimulating the cholinergic receptors muscarinic and nicotinic system, reduces pain it may be an alternative for controlling the pain processes.

Light-harvesting energy transfer and subsequent electron transfer of cationic porphyrin complexes on clay surfaces. A novel energy-transfer system involving nonaggregated cationic porphyrins adsorbed on an anionic-type clay surface and the electron-transfer reaction that occurs after light harvesting are described. In the clay-porphyrin complexes, photochemical energy transfer from excited singlet zinc porphyrins to free-base porphyrins proceeds. The photochemical electron-transfer reaction from an electron donor in solution (hydroquinone) to the adsorbed porphyrin in the excited singlet state was also examined. Because the electron-transfer rate from the hydroquinone to the excited singlet free-base porphyrin is larger than that to the excited singlet zinc porphyrin, we conclude that the energy transfer accelerates the overall electron-transfer reaction.

FR7: Hall of Heroes (1e/2e) The Forgotten Realms fantasy world has been the site of many great adventures and home to countless valiant heroes and infamous villains. Many a tale of derring-do has been recounted in novels like the Moonshae trilogy by Douglas Niles, The Crystal Shard by R. A. Salvatore, Azure Bonds by Kate Novak and Jeff Grubb, and Spellfire by Ed Greenwood (creator of the Forgotten Realms fantasy world). Until now, the characters from these Forgotten Realms novels have been unavailable to AD&D game players. But no longer. Hall of Heroes provides complete histories and AD&D game statistics for such beloved characters as Elminster, the greatest sage and magic user of the Realms, Tristan Kendrick and Robyn of the Moonshae isles, Wulfgar the Barbarian, Drizzt the dark elf, and Bruenor the dwarf, all from the northern reaches, plus Alias and Dragonbait, Shandril and Narm Tamaraith, and many more. But the Hall of Heroes is more that a listing of game statistics: It's also a sourcebook describing (among other things) artificial and magical lifeforms of the Realms, as well as the lives of the world's elves, dwarves, lizardmen, and exotic creatures. Whatever your interest in the Forgotten Realms campaign setting, you'll find much to entertain and intrigue you in this volume. Product History FR7: "Hall of Heroes" (1989), by Steve Perrin, is the sixth book in the "FR" series of sourcebooks for the Forgotten Realms. It was released in February 1989. Origins (I): The FR Series. In 1987 and 1988, TSR published a half-dozen "FR" sourcebooks for the Forgotten Realms, most of them geographical. It was a phenomenal amount of support for a setting, matched only by TSR's concurrent support of the Known World with its line of Gazetteeers (1987-1991). "Hall of Heroes" was something new for the "FR" series: a book of NPCs. Origins (II): The New Edition. 1989 brought big changes for the AD&D game, in the form of the second edition rules (1989). The new Player's Handbook (1989) was published simultaneously with "Hall of Heroes" in February 1989, while the cover of "Hall of Heroes" proudly proclaimed that it was one of the first supplements for "AD&D 2nd Edition". Except it wasn't. Within "Hall of Heroes" you'll find barbarians, cavaliers, and non-specialized magic-users. It's pretty much a first edition book with second edition trade dress. But that was how TSR rolled in late 1988 and early 1989: their books randomly mixed promises of 1st and 2nd edition support, irrespective of what was inside, because the changeover just wasn't seen as that big of a deal. Origins (III): The Realms Novels. The Forgotten Realms line wasn't just TSR's best supported roleplaying line, it was also TSR's best supported fiction line; novels had been an integral part of the publication plans since the release of Darkwalker on Moonshae (1987), two months in advance of the Forgotten Realms Campaign Set (1987) itself. By February 1989, just less than two years later, there were seven Realms novels. Douglas Niles had just completed his Moonshae trilogy of Darkwalker on Moonshae (1987), Black Wizards (1988), and Darkwell (1989), while R.A. Salvatore was two books into his Icewind Dale trilogy, with The Crystal Shard (1988) and Streams of Silver (1989). Finally, TSR had published two books that were intended to be standalone: Spellfire (1988) by Ed Greenwood and Azure Bonds (1988) by Kate Novak and Jeff Grubb. These novels offered a horde of Realmslore every bit as valuable as that contained in the Realms' adventures and sourcebooks … if only the gap between fiction and roleplaying could be bridged. Origins (IV): Bridging the Books. There had already been some cooperation between TSR's fiction and roleplaying lines, going back to the linked Dragonlance Chronicles novels (1984-1985) and adventures (1984-1986). The Forgotten Realms line had also seen some crossover; Moonshae creator Douglas Niles wrote FR2: "Moonshae" (1987) while Jennell Jaquays incorporated some of R.A. Salvatore's northern worldbuilding into FR5: "The Savage Frontier" (1988). That cooperation would increase in 1989, thanks in large part to the Forgotten Realms' new fiction line editor, James Lowder. Generally, Lowder did his best to ensure that authors like Jeff Grubb, Kate Novak, and R.A. Salvatore were as involved as possible with "Hall of Heroes". He then worked with Ed Greenwood, who offered some additional notes on the content. At the time, Lowder was also coordinating the Realms' biggest ever fictional event, the Avatar trilogy (1989). Together James Lowder and Scott Ciencin — the author of the first two Avatar books, Shadowdale (1989) and Tantras (1989) — were able to link "Hall of Heroes" to the Realms' next/i> novels. They created detailed backstories for the Avatar protagonists, even as the novels were being written. These bios were then passed on to writer David Martin so that the stars of the Realms' would be among the heroes of "Hall". Two years after the Realms' inception at TSR, it was more than ever the work of many hands. Increasingly, those hands were spread across the world: with Ciencin in Florida, Salvatore in Massachusetts, and Greenwood in Canada. James Lowder's work as fiction line editor and coordinator showed how those diverse creators could still work together to create a coherent world. Origins (V): The Missing Book. James Lowder's coordination of "Hall of Heroes" ensured that it supported the first ten Realms novels … but there's actually one more in there. David "Zeb" Cook had proposed a Forgotten Realms novel set in his own Kara-Tur. It was under consideration by Jean Black, the head of TSR's fiction department, and presumably would have been the eleventh Forgotten Realms novel. Its characters are all included in "Hall of Heroes". Unfortunately, that novel never came to be. It was probably dropped when Mary Kirchoff took over the fiction department — a common fate for proposals caught between editorial regimes. Cook would eventually get to write a Kara-Tur-related novel when he led off the Empires trilogy with Horselords (1990), though it would be Troy Denning who actually passed into the east with the second novel, Draognwall (1990). Meanwhile, Cook's original characters would return in two adventures, OA6: "Ronin Challenge" (1990) and OA7: "Test of the Samurai" (1990). Origins (VI): The Power of NPCs. NPCs were an integral element of the Forgotten Realms. This was obvious from the first time that Elminster himself wrote about "Pages from the Mages" in Dragon #62 (June 1982). The Realms novels only increased the importance of non-player characters. In later years, the Realms would increasingly acrue a reputation for being a setting full of powerful and important NPCs. "Hall of Heroes", with its incorporation of fiction heroes into the roleplaying line, was a vital stepping stone on that path. Expanding D&D. "Hall of Heroes" features the first rules for "spellfire", a unique magic system in the Forgotten Realms. They'd be repeated in Heroes' Lorebook (1996) and Volo's Guide to All Things Magical (1996). NPCs of Note.Elminster is the only "major character" found in "Hall of Heroes" who isn't a major protagonist in a novel, though he was prominent in Spellfire. The rest of the major characters in "Hall of Heroes" all link more directly to fiction: "Hall of Heroes" also includes shorter writeups of many "minor characters", which include lesser characters from the novels, notables of Cormyr, the Dalelands, and Waterdeep, and a group of heroes then known as the Five Sisters, who would feature more prominently in FOR6: The Seven Sisters (1995) NPCs of Note: Organization. Adventuring Companies were always important in the Realms, from Ed Greenwood's earliest D&D adventures. Two are featured in Hall of Heroes. The Knights of Myth Drannor are based on one of Greenwood's long-lived roleplaying groups, though their history here doesn't necessarily match the one revealed much later by Ed Greenwood in the Knights of Myth Drannor trilogy (2006-2008). The Company of Eight was a company created by Scott Haring for FR3: "Empires of the Sand" (1988). About the Creators. "Hall of Heroes" was edited by Scott Bowles with some coordination by James Lowder. Jeff Grubb, Kate Novak, and R.A. Salvatore each wrote their own NPCs. David E. Martin wrote the Avatar NPCs, Mike Pondsmith wrote the Kara-Tur NPCs, and Steve Perrin wrote the Moonshae NPCs (and many others). Bruce Nesmith finished the major novel characters with the Spellfire NPCs (and many other Daleland personas). James Lowder filled in many of the gaps, from King Azoun IV to Olive Ruskettle. About the Product Historian The history of this product was researched and written by Shannon Appelcline, the editor-in-chief of RPGnet and the author of Designers & Dragons - a history of the roleplaying industry told one company at a time. Please feel free to mail corrections, comments, and additions to shannon.appelcline@gmail.com. These products were created by scanning an original printed edition. Most older books are in scanned image format because original digital layout files never existed or were no longer available from the publisher. For PDF download editions, each page has been run through Optical Character Recognition (OCR) software to attempt to decipher the printed text. The result of this OCR process is placed invisibly behind the picture of each scanned page, to allow for text searching. However, any text in a given book set on a graphical background or in handwritten fonts would most likely not be picked up by the OCR software, and is therefore not searchable. Also, a few larger books may be resampled to fit into the system, and may not have this searchable text background. For printed books, we have performed high-resolution scans of an original hardcopy of the book. We essentially digitally re-master the book. Unfortunately, the resulting quality of these books is not as high. It's the problem of making a copy of a copy. The text is fine for reading, but illustration work starts to run dark, pixellating and/or losing shades of grey. Moiré patterns may develop in photos. We mark clearly which print titles come from scanned image books so that you can make an informed purchase decision about the quality of what you will receive. Original electronic format These ebooks were created from the original electronic layout files, and therefore are fully text searchable. Also, their file size tends to be smaller than scanned image books. Most newer books are in the original electronic format. Both download and print editions of such books should be high quality. File Information Watermarked PDF Adobe DRM-protected PDF These eBooks are protected by Adobe's Digital Rights Management (DRM) technology. To use them, you must activate your Adobe Reader software. Click here for more details. Watermarked PDF These eBooks are digitally watermarked to signify that you are the owner. A small message is added to the bottom of each page of the document containing your name and the order number of your eBook purchase. Warning: If any books bearing your information are found being distributed illegally, then your account will be suspended and legal action may be taken against you.

#!/usr/bin/env python # -*- coding: utf-8 -*- import numpy as np from jsk_topic_tools import ConnectionBasedTransport import rospy from sensor_msgs.msg import Image import cv_bridge class MaskImageToLabel(ConnectionBasedTransport): def __init__(self): super(MaskImageToLabel, self).__init__() self._pub = self.advertise('~output', Image, queue_size=1) def subscribe(self): self._sub = rospy.Subscriber('~input', Image, self._apply) def unsubscribe(self): self._sub.unregister() def _apply(self, msg): bridge = cv_bridge.CvBridge() mask = bridge.imgmsg_to_cv2(msg, desired_encoding='mono8') if mask.size == 0: rospy.logdebug('Skipping empty image') return label = np.zeros(mask.shape, dtype=np.int32) label[mask == 0] = 0 label[mask == 255] = 1 label_msg = bridge.cv2_to_imgmsg(label, encoding='32SC1') label_msg.header = msg.header self._pub.publish(label_msg) if __name__ == '__main__': rospy.init_node('mask_image_to_label') mask2label = MaskImageToLabel() rospy.spin()

Mowtowr-e Mir Huti Naruyi Mowtowr-e Mir Huti Naruyi (, also Romanized as Mowtowr-e Mīr Hūtī Nārūyī) is a village in Jolgeh-ye Chah Hashem Rural District, Jolgeh-ye Chah Hashem District, Dalgan County, Sistan and Baluchestan Province, Iran. At the 2006 census, its population was 98, in 20 families. References Category:Populated places in Dalgan County

Optimization of large volume injection-programmable temperature vaporization-gas chromatography-mass spectrometry analysis for the determination of estrogenic compounds in environmental samples. Large volume injection-programmable temperature vaporization-gas chromatography-mass spectrometry (LVI-PTV-GC-MS) was optimized for the determination of estrone (E1), 17β-estradiol (E2), 17α-ethynyl estradiol (EE2), mestranol (MeEE2) and estriol (E3) for their determination in environmental samples (estuarine water, wastewater, fish bile and fish homogenate) after derivatization with 25 μL (BSTFA+1% TMCS) and 125 μL of pyridine. Experimental designs such as Plackett-Burman (PBD) and central composite designs (CCDs) were used to optimize the LVI-PTV variables (cryo-focusing temperature, vent time, vent flow, vent pressure, injection volume, purge flow to split vent, splitless time and injection speed). Optimized conditions were as follows: 45 μL of n-hexane extract are injected at 60°C and 6 μL/s with a vent flow and a vent pressure of 50 mL/min and 7.7 psi, respectively, during 5 min; then the split valve is closed for 1.5 min and afterwards the injector is cleaned at 100 mL/min before the next injection. The method was applied to the determination of estrogenic compounds in environmental samples such as estuarine water, wastewater, and fish homogenate and bile. Limits of detection (0.04-0.15 ng/L for water samples, 0.04-0.67 ng/g for fish bile and 0.1-7.5 ng for fish homogenate) obtained were approx. ten times lower than those obtained by means of a common split/splitless inlet.

Collingwood Magpies AFL Club Be part of the action with Emirates Live Collingwood Magpies have become one of the best AFL teams in Australia. They boast of 15VFL/AFL premierships champion titles and have competed in more grand finals that any other club in the history of AFL sport, giving themselves quite a name in the Victoria state. Surround yourself in magpies and give your support to Collingwood this coming season. Considering their previous accomplishments, it's nearly a 100% guarantee you won’t be disappointed.

--- abstract: 'The current article shall contribute to understanding the classical analogue of the minimal photon sector in the Lorentz-violating Standard-Model Extension (SME). It is supposed to complement all studies performed on classical point-particle equivalents of SME fermions. The classical analogue of a photon is not a massive particle being described by a usual equation of motion, but a geometric ray underlying the eikonal equation. The first part of the paper will set up the necessary tools to understand this correspondence for interesting cases of the minimal SME photon sector. In conventional optics the eikonal equation follows from an action principle, which is demonstrated to work in most (but not all) Lorentz-violating cases as well. The integrands of the action functional correspond to Finsler structures, which establishes the connection to Finsler geometry. The second part of the article treats Lorentz-violating light rays in a weak gravitational background by implementing the principle of minimal coupling. Thereby it is shown how Lorentz violation in the photon sector can be constrained by measurements of light bending at massive bodies such as the Sun. The phenomenological studies are based on the currently running ESA mission GAIA and the planned NASA/ESA mission LATOR. The final part of the paper discusses certain aspects of explicit Lorentz violation in gravity based on the setting of Finsler geometry.' author: - 'M. Schreck' title: 'Eikonal approximation, Finsler structures, and implications for Lorentz-violating photons in weak gravitational fields' --- Introduction ============ During the past 15 years plenty of progress has been made in understanding [*CPT*]{}- and Lorentz violation and its possible implications on physics from both a theoretical and a phenomenological point of view. This was made possible by establishing the Standard-Model Extension (SME) in 1998 [@Colladay:1998fq] and by the subsequent tireless work of people in our community eager to study imprints of Planck-scale physics detectable by experiments operating at much smaller energies. The SME is a powerful framework incorporating all Lorentz-violating operators into the Standard Model of elementary particles and General Relativity. It neither modifies the gauge structure of the Standard Model nor does it introduce new particles. The power-counting renormalizable contributions of the SME are grouped into its minimal part where the remaining higher-order operators comprise the nonminimal SME [@Kostelecky:2009zp; @Kostelecky:2011gq; @Kostelecky:2013rta]. This framework allows for astounding experimental tests of Lorentz invariance where even presently some experiments reach a sensitivity of the Planck scale square (see [@Kostelecky:2008ts] for a yearly updated compilation of experimental contraints on Lorentz-violating coefficients). Since Lorentz violation implies [*CPT*]{}-violation according to a theorem by Greenberg [@Greenberg:2002uu], the Standard-Model Extension involves all [*CPT*]{}-odd operators as a subset. Note that Lorentz violation has been predicted by various prototypes of fundamental theories such as string theory [@Kostelecky:1988zi; @Kostelecky:1991ak; @Kostelecky:1994rn], loop quantum gravity [@Gambini:1998it; @Bojowald:2004bb], noncommutative spacetime [@AmelinoCamelia:1999pm; @Carroll:2001ws], spacetime foam [@Klinkhamer:2003ec; @Bernadotte:2006ya; @Hossenfelder:2014hha], and models with nontrivial spacetime topology [@Klinkhamer:1998fa; @Klinkhamer:1999zh]. In the recent past profound studies of modified quantum field theories based on the SME were performed at tree-level and including quantum corrections. The result of these studies is that most sectors are free of any inconsistencies [@Kostelecky:2000mm; @oai:arXiv.org:hep-ph/0101087; @Casana-etal2009; @Casana-etal2010; @Klinkhamer:2010zs; @Schreck:2011ai; @Cambiaso:2012vb; @Colladay:2014dua; @Maniatis:2014xja; @Schreck:2013gma; @Schreck:2013kja; @Cambiaso:2014eba; @Schreck:2014qka; @Colladay:2014dua; @Albayrak:2015ewa]. Furthermore the SME was explicitly shown to be renormalizable at one loop [@Kostelecky:2001jc; @Colladay:2006rk; @Colladay:2007aj; @Colladay:2009rb] where latest computations have demonstrated renormalizability of the modified quantum electrodynamics [@Santos:2015koa] and the pure Yang-Mills sector [@Santos:2014lfa] at infinite-loop order using algebraic techniques. Therefore as long as the SME is restricted to Minkowski spacetime, it seems to be a reasonable, well-behaved, and model-independent test framework for Planck-scale physics. The gravitational sector of the SME was constructed in the seminal article [@Kostelecky:2003fs]. In the aftermath, studies on its theory and phenomenology were performed in a successive series of papers [@Bailey:2006fd; @Kostelecky:2007kx; @Kostelecky:2008in; @Bailey:2009me; @Kostelecky:2010ze; @Tasson:2010nr; @Tasson:2012nx; @Tasson:2012au; @Bonder:2013sca; @Bonder:2015maa] with recent investigations of even nonminimal operators in short-range gravity tests [@Bailey:2014bta; @Long:2014swa]. One of the most important theoretical results of [@Kostelecky:2003fs] is a no-go theorem stating that explicit Lorentz violation is incompatible with the geometric framework of General Relativity, which is Riemannian geometry. Considering Lorentz-violating matter in a gravitational background results in modified conservation laws of the energy-momentum tensor based on Noether’s theorem. However Lorentz violation does [*a priori*]{} not modify the geometrical base such as the Bianchi identities of the Riemann curvature tensor. Due to the Einstein equations the second Bianchi identity is tightly bound to the conservation of energy-momentum, which is then incommensurate with the modified matter sector. A possibility of circumventing this clash is to perform phenomenological studies in theories resting on spontaneous Lorentz violation. This means that a Lorentz-violating background field arises dynamically as the vacuum expectation value of a vector or tensor field. Such models have been studied since the early 1990s [@Kostelecky:1988zi; @Kostelecky:1989jp; @Kostelecky:1989jw; @Bluhm:2008yt; @Hernaski:2014jsa; @Bluhm:2014oua] (even before the SME existed) and they can be considered as one of the motivations that lead to the construction of the SME. The crucial point within models of spontaneous Lorentz violation is to take into account the Nambu-Goldstone modes that are linked to the symmetry breaking. This can lead to arduous perturbative calculations within such a theory. For these reasons it would be preferable to have a setup available that allows for incorporating explicit Lorentz violation into a curved background without possible tensions with the underlying geometrical properties. A suggestion was already given in [@Kostelecky:2003fs] along the same lines as the no-go theorem: introducing an alternative geometrical framework that can include preferred directions naturally. Such an extension of Riemannian geometry has been known in the mathematics community for almost 100 years. It is named Finsler geometry in reference to the famous mathematician Finsler who studied generalized path length functionals in his Ph.D. thesis [@Finsler:1918; @Cartan:1933] (cf. [@Bao:2000] for a comprehensive mathematical overview on the subject). Finsler geometry has been applied to various fields of physics [@Antonelli:1993]. In the context of the Standard-Model Extension it found its use just a couple of years ago when it was shown that the minimal Lorentz-violating fermion sector can be mapped to classical-particle descriptions [@Kostelecky:2010hs; @Kostelecky:2011qz; @Kostelecky:2012ac; @Colladay:2012rv; @Russell:2015gwa]. The corresponding Lagrange functions are closely linked to Finsler structures, i.e., generalized path length functionals. Recently a nonminimal case was studied [@Schreck:2014hga] as well as classical-particle trajectories in electromagnetic fields and modified spin precession based on an isotropic set of minimal fermion coefficients [@Schreck:2014ama]. In [@Silva:2013xba] a particular class of Finsler spaces known as bipartite is investigated closer from a physics point of view and [@Foster:2015yta] suggests classical-mechanics systems that are linked to three-dimensional versions of Finsler $b$ space [@Kostelecky:2011qz]. In a very recent paper [@Colladay:2015wra] $b$ space is discussed from a mathematical point of view. Its indicatrix (surface of constant value of the Finsler structure) is a two-valued deformation from a sphere that is characterized by singularities with ambiguous derivatives. Considering the indicatrix as an algebraic variety, the Hironaka theorem says that such singularities can be removed [@Hironaka:1964]. In [@Colladay:2015wra] a coordinate transformation was found, which allows to remove the singular sets and to glue the remaining parts together appropriately. This results in a well-defined mathematical description of $b$ space that can be used for future physical investigations. The goals of the current article are threefold. First, analogous classical equivalents for the minimal [*CPT*]{}-even photon sector of the SME shall be found. Second, with these equivalents at hand we intend to study phenomenological aspects of Lorentz-violating photons in weak gravitational fields. Last but not least we would like to understand various consequences of this approach on the base of Finsler geometry. The procedures to be developed will differ extensively from the SME fermion counterparts. The paper is organized as follows. In the Lorentz-violating framework, which all investigations are based on, is introduced. A brief review on Finsler geometry and Finsler structures in the SME fermion sector is given in , followed by an explanation of the method to constructing Finsler structures in the photon sector. In that section we investigate different cases that are the most interesting ones from a physics point of view. In the geometric-optics approximation photons are described by the eikonal equation, which forms the cornerstone of . It is demonstrated how the Finsler structures obtained are linked to the eikonal equation for the different sectors analyzed in the previous section. Since the isotropic modification of the [*CPT*]{}-even sector can be considered to be the most important one, all forthcoming studies will be based on the latter. Section \[sec:gravitational-backgrounds\] is dedicated to investigating the isotropic eikonal equation in a weak gravitational background. We develop a phenomenological framework to study light bending at massive bodies within such a theory. In this context prospects are given on detecting isotropic Lorentz violation of photons propagating in a gravitational background. This is carried out for two space-based missions: GAIA and LATOR. The final part of the paper is more theoretical. In the modified conservation law of the energy-momentum tensor is investigated, interpreting the results from the point of view of explicit versus spontaneous Lorentz violation. Last but not least, in we examine the properties of the isotropic spacetime that has been subject to the studies in from a Finsler-geometric point of view. The most important findings in total are concluded on and discussed in . Essential calculational details can be found in Appx. \[sec:lagrangians-massive-photons\] to \[sec:eikonal-equation-inhomogeneous-anisotropic\]. Throughout the article natural units with $\hbar=c=1$ are chosen unless otherwise stated. Construction of classical Lagrangians {#sec:construction-classical-lagrangians} ===================================== The base of the current article is formed by the minimal SME photon sector whose action $S_{\upgamma}$ is comprised of [*CPT*]{}-even modified Maxwell (mM) [@Colladay:1998fq; @Kostelecky:2002hh; @BaileyKostelecky2004] theory and [*CPT*]{}-odd Maxwell-Chern-Simons (MCS) theory [@Carroll:1989vb; @Colladay:1998fq; @Kostelecky:2002hh; @BaileyKostelecky2004]: \[eq:action-modified-maxwell-theory\] $$\begin{aligned} S_{\upgamma}&=\int_{\mathbb{R}^4}\mathrm{d}^4x\,\left[\mathcal{L}_\text{mM}(x)+\mathcal{L}_{\mathrm{MCS}}(x)+\mathcal{L}_{\mathrm{mass}}(x)\right]\,, \displaybreak[0]\\[2ex] \label{eq:lagrange-density-modmax} \mathcal{L}_\text{mM}(x)&=-\frac{1}{4}\,\eta^{\mu\rho}\,\eta^{\nu\sigma}\,F_{\mu\nu}(x)F_{\rho\sigma}(x)-\frac{1}{4}\,(k_F)^{\mu\nu\varrho\sigma}\,F_{\mu\nu}(x)F_{\varrho\sigma}(x)\,, \displaybreak[0]\\[2ex] \label{eq:lagrange-density-mcs} \mathcal{L}_{\mathrm{MCS}}(x)&=\frac{m_{\scriptscriptstyle{\mathrm{CS}}}}{2}(k_{AF})^{\kappa}\varepsilon_{\kappa\lambda\mu\nu}A^{\lambda}(x)F^{\mu\nu}(x)\,, \displaybreak[0]\\[2ex] \mathcal{L}_{\mathrm{mass}}(x)&=m_{\upgamma}^2A_{\mu}(x)A^{\mu}(x)\,.\end{aligned}$$ Here $F_{\mu\nu}(x)\equiv \partial_{\mu}A_{\nu}(x)-\partial_{\nu}A_{\mu}(x)$ is the electromagnetic field strength tensor that involves the *U*(1) gauge field $A_{\mu}(x)$. The fields are defined on Minkowski spacetime with metric $(\eta_{\mu\nu})=\mathrm{diag}(1,-1,-1,-1)$. The totally antisymmetric Levi-Civita symbol in four spacetime dimensions is denoted as $\varepsilon^{\mu\nu\varrho\sigma}$ with $\varepsilon^{0123}=1$. The controlling coefficients characteristic for the framework considered are comprised in the fourth-rank observer tensor $(k_F)^{\mu\nu\varrho\sigma}$ and the observer vector $(k_{AF})^{\kappa}$. Both have dimensionless components and they do not transform covariantly with respect to particle Lorentz transformations, which renders this theory explicitly Lorentz-violating. The field operator of modified Maxwell theory is of dimension four, whereas the operator of MCS theory has mass dimension three. Therefore MCS theory involves the Chern-Simons mass scale $m_{\scriptscriptstyle{\mathrm{CS}}}$ for dimensional consistency. It is well-known that a photon mass term encoded in $\mathcal{L}_{\mathrm{mass}}$ (with the photon mass $m_{\upgamma}$) violates *U*(1) gauge invariance. It has been introduced here for certain purposes that will be explained below, but for most occasions $m_{\upgamma}$ will be set to zero. Anyhow in [@Colladay:2014dua] it was demonstrated that certain birefringent cases of modified Maxwell theory require a nonvanishing photon mass (at least in intermediate calculations) to have a consistent Gupta-Bleuler quantization. Finally, a gauge fixing term will be omitted in the action, since all considerations will be carried out at the classical level. Classical Lagrangians and Finsler structures {#sec:classical-lagrangians-finsler-structures} -------------------------------------------- The major goal is to understand how Lorentz-violating photons can be described in the context of gravity. Since Einstein’s relativity is a classical theory, it is reasonable to obtain a classical analogue of the quantum field theory based on the action of . With such an analogue at hand it should be possible to study how an explicitly Lorentz-violating theory of gravity could be constructed consistently. As an introduction to the topic the mapping procedure of the SME fermion sector to a classical point-particle description [@Kostelecky:2010hs] shall be reviewed. From a quantum theoretical point of view a particle can be understood as a suitable superposition of free-field solutions with dispersion relation $$\label{eq:dispersion-relation-general} f(p_{\mu},m_{\psi},k_x)=0\,,\quad (p_{\mu})=\begin{pmatrix} p_0 \\ \mathbf{p} \\ \end{pmatrix}\,,$$ such that its probability density is nonzero in a localized region and drops off to zero sufficiently fast outside. Here $p_0$ is the particle energy, $\mathbf{p}$ its three-momentum, $m_{\psi}$ the fermion mass, and $k_x$ denotes a particular set of Lorentz-violating coefficients where $x$ represents a Lorentz index structure. The physical propagation velocity of such a wave packet is the group velocity $$\mathbf{v}_{\mathrm{gr}}\equiv \frac{\partial p_0}{\partial\mathbf{p}}\,.$$ A classical, relativistic pointlike particle is assumed to propagate with four-velocity $u^{\mu}=\gamma(1,\mathbf{v})$ where $\mathbf{v}$ is the three-velocity. To map the wave packet to such a classical particle, it makes sense to identify the group velocity components with the appropriate spatial four-velocity components: $$\label{eq:group-velocity-correspondence} \mathbf{v}_{\mathrm{gr}}\overset{!}{=} -\frac{\mathbf{u}}{u^0}\,.$$ The minus sign has its origin in the different position of the spatial index on both sides of the equation. Since the physics of the classical particle rests on a Lagrange function $L=L(u^0,\mathbf{u})$, its construction is of paramount importance. If the Lagrange function is positive homogeneous of degree one, i.e., $L(\lambda u^0,\lambda\mathbf{u})=\lambda L(u^0,\mathbf{u})$ for $\lambda>0$, the action is parameterization-invariant. In this case the physics does not depend on the way how the particle trajectory is parameterized, which is a very reasonable property to have. Positive homogeneity gives the following condition on the Lagrange density according to Euler’s theorem [@Bao:2000]: $$\label{eq:lagrange-function} L=-p_{\mu}u^{\mu}\,,\quad p_{\mu}=-\frac{\partial L}{\partial u^{\mu}}\,,$$ with the conjugate momentum $p_{\mu}$. The latter is identified with the momentum that appears in the quantum theoretical dispersion relation of . The global minus sign in its definition has been introduced such that the nonrelativistic kinetic energy is positive. Now Eqs. (\[eq:dispersion-relation-general\]), (\[eq:group-velocity-correspondence\]), and (\[eq:lagrange-function\]) comprise a set of five conditions that shall be used to determine $p_{\mu}$ and $L$. Hence all four-momentum components and the Lagrange function are supposed to be solely expressed in terms of four-velocity components. The Lagrange functions corresponding to the standard fermion dispersion law $p_0^2-\mathbf{p}^2-m_{\psi}^2=0$ read $L=\pm m_{\psi}\sqrt{(u^0)^2-\mathbf{u}^2}$. The two signs are the classical counterparts of the particle-antiparticle solutions at the level of quantum field theory. It can be checked that the five equations above are fulfilled for this choice of $L$. The latter can also be written in the form $L=\pm m_{\psi}\sqrt{r_{\mu\nu}u^{\mu}u^{\nu}}$ with $r_{\mu\nu}$ known as the intrinsic metric. This metric is essential to determine lengths of vectors and angles enclosed by vectors. In the particular case considered it corresponds to the (indefinite) Minkowski metric: $r_{\mu\nu}=\eta_{\mu\nu}$. This is not surprising, since the starting point to obtaining the Lagrange function was a field theory defined in Minkowski spacetime. By a Wick rotation the Lagrange function is related to a new function $F$ based on a positive definite intrinsic metric: $$F(y)\equiv F(\mathbf{y},y^4)\equiv \frac{\mathrm{i}}{m_{\psi}}L(\mathrm{i}y^4,\mathbf{y})=\sqrt{r_{ij}y^iy^j}\,,\quad r_{ij}=\mathrm{diag}(1,1,1,1)_{ij}\,.$$ Promoting $r_{ij}$ to an arbitrary position-dependent metric $r_{ij}(x)$, the function $F$ becomes dependent on $x$: $F(y)\mapsto F(x,y)$. It can then be interpreted as the integrand of a path length functional of a Riemannian manifold $M$ where $y\in T_xM$. A Finsler structure is a generalization of that obeying the following properties: - $F(x,y)>0$, - $F(x,y)\in C^{\infty}$ for all $y\in T_xM\setminus \{\mathrm{slits}\}$, - positive homogeneity in $y$, i.e., $F(x,\lambda y)=\lambda F(x,y)$ for $\lambda>0$, and - the derived metric (Finsler metric) $$g_{ij}\equiv \frac{1}{2}\frac{\partial^2 F^2}{\partial y^i\partial y^j}\,,$$ is positive definite. Prominent examples for Finsler structures that are outside the scope of Riemannian geometry are Randers structures, $F(y)=\alpha+\beta$, and Kropina structures, $F(y)=\alpha^2/\beta$, with $\alpha=\sqrt{a_{ij}y^iy^j}$ and $\beta=b_iy^i$ where $a_{ij}$ is a Riemannian metric and $b_i$ a one-form. There are certain theorems available to classify Finsler structures using various kinds of torsions. The most important one is the Cartan torsion $C_{ijk}$, which is given by [@Bao:2004] $$\label{eq:cartan-torsion} C_{ijk}\equiv \frac{1}{2}\frac{\partial g_{ij}}{\partial y^k}=\frac{1}{4}\frac{\partial^3F^2}{\partial y^i\partial y^j\partial y^k}\,.$$ In some books $C_{ijk}$ is defined with an additional prefactor $F$ (see, e.g., [@Bao:2000]). The mean Cartan torsion reads as follows: $$\label{eq:mean-cartan-torsion} \mathbf{I}\equiv I_iy^i\,,\quad I_i\equiv g^{jk}C_{ijk}\,,\quad (g^{ij})\equiv (g_{ij})^{-1}\,,$$ with the inverse derived metric $g^{ij}$. Deicke’s theorem says that a Finsler space is Riemannian if and only if $\mathbf{I}$ vanishes [@Deicke:1953]. The Matsumoto torsion provides a further set of quantities that are very useful to classify Finsler structures: $$\label{eq:matsumoto-torsion} M_{ijk}\equiv C_{ijk}-\frac{1}{n+1}(I_ih_{jk}+I_jh_{ik}+I_kh_{ij})\,,\quad h_{ij}\equiv F \frac{\partial^2F}{\partial y^i\partial y^j}\,.$$ Here $n$ is the dimension of the Finsler structure considered [@Bao:2004]. According to the Matsumoto-Hōjō theorem a Finsler structure is either of Randers or Kropina type if and only if the Matsumoto torsion is equal to zero [@Matsumoto:1978]. These theorems will be used frequently throughout the paper to classify Finsler structures encountered. According to the rules recalled at the beginning of the current section classical Lagrange functions of the SME fermion sector were derived in [@Kostelecky:2010hs; @Colladay:2012rv; @Schreck:2014hga; @Schreck:2014ama; @Russell:2015gwa]. In the articles [@Colladay:2012rv; @Kostelecky:2011qz; @Kostelecky:2012ac; @Schreck:2014hga] their corresponding Finsler structures were examined. In this paper analogous investigations shall be performed for the minimal SME photon sector based on the action of . It will become evident that the possible techniques used differ from the procedures adopted for the fermion sector. Maxwell-Chern-Simons theory --------------------------- In the current section the [*CPT*]{}-even photon sector components $(k_F)^{\mu\nu\varrho\sigma}$ in will be set to zero restricting our considerations to the MCS term of only. Furthermore the photon mass $m_{\upgamma}$ will be set to zero as well. In the seminal article [@Carroll:1989vb] the magnitude of $m_{\scriptscriptstyle{\mathrm{CS}}}(k_{AF})^{\kappa}$ was constrained tightly due to the absence of astrophysical birefringence. A collection of all constraints on components of $m_{\scriptscriptstyle{\mathrm{CS}}}(k_{AF})^{\kappa}$ can be found in the data tables [@Kostelecky:2008ts]. In spite of the tight bounds, MCS theory is very interesting from a theoretical point of view. The structure of the quantum field theory based on MCS theory is quite involved, which was shown by extensive investigations carried out in [@oai:arXiv.org:hep-ph/0101087].[^1] The smoking-gun results of the latter reference are that MCS theory is well-behaved as long as the preferred spacetime direction $(k_{AF})^{\kappa}$ is spacelike. For timelike $(k_{AF})^{\kappa}$ issues with either microcausality or unitarity arise, though. Interestingly this behavior mirrors in the classical Finsler structure of MCS theory that will be derived as follows. First of all spacelike MCS theory shall be considered. The modified field equations in momentum space read as follows [@Colladay:1998fq]: $$\begin{aligned} M^{\alpha\delta}(p)A_{\delta}&=0\,, \\[2ex] M^{\alpha\delta}(p)&=\eta^{\alpha\delta}k^2-k^{\alpha}k^{\delta}-2\mathrm{i}m_{\scriptscriptstyle{\mathrm{CS}}}(k_{AF})_{\beta}\varepsilon^{\alpha\beta\gamma\delta}k_{\gamma}\,,\end{aligned}$$ where $k_{\mu}$ is the four-momentum to be distinguished from the four-momentum $p_{\mu}$ used for fermions. Imposing Lorenz gauge $k^{\delta}A_{\delta}=0$, the condition of a vanishing determinant of $M$ results in $$k^4+4m_{\scriptscriptstyle{\mathrm{CS}}}^2\left[k^2(k_{AF})^2-(k\cdot k_{AF})^2\right]=0\,,$$ leading to the following dispersion relations: $$\omega_{1,2}=\sqrt{\mathbf{k}^2+2m_{\scriptscriptstyle{\mathrm{CS}}}^2(\mathbf{k}_{AF})^2\pm 2m_{\scriptscriptstyle{\mathrm{CS}}}\sqrt{m_{\scriptscriptstyle{\mathrm{CS}}}^2(\mathbf{k}_{AF})^4+(\mathbf{k}\cdot \mathbf{k}_{AF})^2}}\,.$$ Here the spatial momentum $\mathbf{k}$ is not to be confused with the spatial part $\mathbf{k}_{AF}$ of the MCS vector. Following the procedure outlined in leads to the Lagrange function $$\label{eq:lagrange-function-mcs-theory} L|_{\mathrm{MCS}}^{\pm}=\pm m_{\scriptscriptstyle{\mathrm{CS}}}\left(\sqrt{-(k_{AF})^2u^2}\pm \sqrt{(k_{AF}\cdot u)^2-(k_{AF})^2u^2}\right)\,.$$ First of all, this result matches the Lagrange function first obtained in [@McGinnis:2014]. For spacelike $k_{AF}$ it corresponds to the Lagrange density of the minimal fermionic $b^{\mu}$ coefficient where here $(k_{AF})^{\mu}$ takes the role of $b^{\mu}$ and the Chern-Simons mass $m_{\scriptscriptstyle{\mathrm{CS}}}$ takes the role of the fermion mass $m_{\psi}$. This is because there exists a correspondence between MCS theory and the fermion theory involving the $b^{\mu}$ coefficient whose Lagrangian has the form $b^{\mu}\overline{\psi}\gamma_5\gamma_{\mu}\psi$. The associated field operator is of dimension three and it is [*CPT*]{}-odd [@Kostelecky:2009zp], which parallels some of the properties of MCS theory. Therefore the Wick-rotated version of can be interpreted as a $b$ space. The form of the Lagrangian of remains the same even for MCS theory with a timelike $k_{AF}$, which can be shown by direct computation. Undoubtedly, issues arise for timelike $k_{AF}$, since in this case the Lagrange function is not a real function any more. A classical Lagrange function is of mass dimension one, which is why is directly proportional to the single mass scale $m_{\scriptscriptstyle{\mathrm{CS}}}$ that appears in this framework. In the limit $m_{\scriptscriptstyle{\mathrm{CS}}}\mapsto 0$ the Lagrange function vanishes, which reveals the challenge in deriving appropriate Lagrange functions corresponding to Lorentz-violating frameworks that do not have a dimensional scale associated to them. This is especially the case for a photon theory based on modified Maxwell theory, which will be discussed as follows. Modified Maxwell theory {#eq:lagrange-function-massive-modmax} ----------------------- In the remainder of the paper the Chern-Simons mass $m_{\scriptscriptstyle{\mathrm{CS}}}$ will be set to zero and the Lagrange density of MCS theory, , will not be taken into account any more. The observer four-tensor $(k_F)^{\mu\nu\varrho\sigma}$ in will be decomposed into contributions involving the Minkowski metric and a $(4\times 4)$ matrix $\widetilde{\kappa}^{\mu\nu}$ according to the nonbirefringent *Ansatz* [@BaileyKostelecky2004; @Altschul:2006zz]$$\label{eq:nonbirefringent-ansatz} (k_F)^{\mu\nu\varrho\sigma}=\frac{1}{2}(\eta^{\mu\varrho}\widetilde{\kappa}^{\nu\sigma}-\eta^{\mu\sigma}\widetilde{\kappa}^{\nu\varrho}-\eta^{\nu\varrho}\widetilde{\kappa}^{\mu\sigma}+\eta^{\nu\sigma}\widetilde{\kappa}^{\mu\varrho})\,.$$ The matrix $\widetilde{\kappa}^{\mu\nu}$ is supposed to be symmetric and traceless. Its particular choice amounts to different Lorentz-violating cases in the minimal, [*CPT*]{}-even photon sector characterized by nonbirefringent photon dispersion laws at first order in the Lorentz-violating coefficients. This means that resulting dispersion relations for the two physical photon polarization states coincide with each other at first order in Lorentz violation. The notation — especially for the controlling coefficients — is mainly based on [@Kostelecky:2002hh]. First of all the photon mass is kept. The equations of motion for the photon field $A_{\mu}$ in momentum space then take the following form [@Colladay:1998fq; @McGinnis:2014]: \[eq:field-equations-modmax\] $$\begin{aligned} M^{\alpha\delta}(k)A_{\delta}&=0\,, \\[2ex] M^{\alpha\delta}(k)&=\eta^{\alpha\delta}(k^2-m_{\upgamma}^2)-k^{\alpha}k^{\delta}-2(k_F)^{\alpha\beta\gamma\delta}k_{\beta}k_{\gamma}\,.\end{aligned}$$ Now different interesting cases of modified Maxwell theory (including a photon mass term) will be examined. The simplest case is undoubtedly the isotropic one, which is characterized by a single controlling coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ and one preferred timelike spacetime direction $\xi^{\mu}$. The matrix $\widetilde{\kappa}^{\mu\nu}$ is then diagonal and it is given as follows: $$\begin{aligned} \label{eq:matrix-kappas-isotropic-case} \widetilde{\kappa}^{\mu\nu}&=2\widetilde{\kappa}_{\mathrm{tr}}\left(\xi^{\mu}\xi^{\nu}-\frac{1}{4}\xi^2\eta^{\mu\nu}\right)=\frac{3}{2}\widetilde{\kappa}_{\mathrm{tr}}\,\mathrm{diag}\left(1,\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)^{\mu\nu}\,, \\[2ex] (\xi^{\mu})&=(1,0,0,0)^T\,.\end{aligned}$$ The dispersion equation, which follows from claiming a vanishing determinant of $M^{\alpha\delta}$ in using Lorenz gauge $k^{\delta}A_{\delta}=0$, results in $$\begin{aligned} \label{eq:dispersion-relation-isotropic-massive} m_{\upgamma}^2&=a^{\mu\nu}k_{\mu}k_{\nu}\,, \\[2ex] a^{\mu\nu}&=\mathrm{diag}\left(1+\widetilde{\kappa}_{\mathrm{tr}},-[1-\widetilde{\kappa}_{\mathrm{tr}}],-[1-\widetilde{\kappa}_{\mathrm{tr}}],-[1-\widetilde{\kappa}_{\mathrm{tr}}]\right)^{\mu\nu}\,.\end{aligned}$$ The next case to be considered is a nonbirefringent, anisotropic one that is characterized by a single (parity-even) controlling coefficient $\widetilde{\kappa}_{e-}^{11}$ and one spacelike direction $\zeta^{\mu}$. Furthermore $\widetilde{\kappa}_{\mathrm{e}-}^{22}=\widetilde{\kappa}_{e-}^{11}$, $\widetilde{\kappa}_{\mathrm{e}-}^{33}=-2\widetilde{\kappa}_{e-}^{11}$ and all remaining ones vanish. The matrix $\widetilde{\kappa}^{\mu\nu}$ for the nonbirefringent *Ansatz* is given as follows: \[eq:kappas-anisotropic-nonbirefringent-case\] $$\begin{aligned} \widetilde{\kappa}^{\mu\nu}&=3\widetilde{\kappa}_{e-}^{11}\left(\zeta^{\mu}\zeta^{\nu}-\frac{1}{4}\zeta^2\eta^{\mu\nu}\right)=\frac{3}{4}\widetilde{\kappa}_{e-}^{11}\,\mathrm{diag}(1,-1,-1,3)^{\mu\nu}\,, \\[2ex] (\zeta^{\mu})&=(0,0,0,1)^T\,.\end{aligned}$$ The latter has a similar structure compared to and it is again diagonal. However its spatial coefficients differ from each other revealing the anisotropy. The modified photon dispersion equation can be written in the same form as for the isotropic case: $$\begin{aligned} \label{eq:dispersion-relation-anisotropic-massive} m_{\upgamma}^2&=b^{\mu\nu}k_{\mu}k_{\nu}\,, \\[2ex] b^{\mu\nu}&=\mathrm{diag}\left(1+\frac{3}{2}\widetilde{\kappa}_{e-}^{11},-\left[1+\frac{3}{2}\widetilde{\kappa}_{e-}^{11}\right],-\left[1+\frac{3}{2}\widetilde{\kappa}_{e-}^{11}\right],-\left[1-\frac{3}{2}\widetilde{\kappa}_{e-}^{11}\right]\right)^{\mu\nu}\,.\end{aligned}$$ The third particular case of modified Maxwell theory to be examined in this context is characterized by three (parity-odd) controlling coefficients $\widetilde{\kappa}_{o+}^{23}$, $\widetilde{\kappa}_{o+}^{31}$, and $\widetilde{\kappa}_{o+}^{12}$ where all remaining ones that are not related by symmetries vanish. Furthermore there are two preferred spacetime directions: a timelike direction $\xi^{\mu}$ and a spacelike one $\zeta^{\mu}$. The matrix $\widetilde{\kappa}^{\mu\nu}$ in the nonbirefrigent *Ansatz* can be cast into $$\begin{aligned} \widetilde{\kappa}^{\mu\nu}&=\frac{1}{2}(\xi^{\mu}\zeta^{\nu}+\zeta^{\mu}\xi^{\nu})-\frac{1}{4}(\xi\cdot\zeta)\eta^{\mu\nu}\,, \\[2ex] \label{eq:four-vectors-parity-odd} (\xi^{\mu})&=(1,0,0,0)^T\,,\quad (\zeta^{\mu})=-2(0,\boldsymbol{\zeta})^T\,,\quad \boldsymbol{\zeta}=(\widetilde{\kappa}_{o+}^{23},\widetilde{\kappa}_{o+}^{31},\widetilde{\kappa}_{o+}^{12})^T\,.\end{aligned}$$ Due to observer Lorentz invariance the coordinate system can be set up such that $\boldsymbol{\zeta}$ points along its third axis. The first photon dispersion equation is quadratic and reads as follows: \[eq:dispersion-relation-parity-odd-massive-1\] $$\begin{aligned} m_{\upgamma}^2&=c^{\mu\nu}k_{\mu}k_{\nu}\,, \\[2ex] c^{\mu\nu}&=\begin{pmatrix} 1 & 0 & 0 & -\mathcal{E} \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -\mathcal{E} & 0 & 0 & -1 \\ \end{pmatrix}^{\mu\nu}\,,\quad \mathcal{E}=\sqrt{(\widetilde{\kappa}_{o+}^{23})^2+(\widetilde{\kappa}_{o+}^{31})^2+(\widetilde{\kappa}_{o+}^{12})^2}\,.\end{aligned}$$ Note that the latter has an equivalent structure to Eqs. (\[eq:dispersion-relation-isotropic-massive\]), (\[eq:dispersion-relation-anisotropic-massive\]). However the second dispersion equation is quartic and it is given by $$\label{eq:dispersion-relation-parity-odd-massive-2} 0=(k^2-m_{\upgamma}^2)^2-(k\cdot\zeta)(k\cdot\xi)(k^2-m_{\upgamma}^2)+\frac{1}{4}\left\{(k\cdot\zeta)^2+\zeta^2[(k\cdot\xi)^2-k^2]\right\}k^2\,.$$ For $m_{\upgamma}=0$ the right-hand side of the latter factorizes into $k^2$ and a quadratic dispersion relation that differs from (for $m_{\upgamma}=0$) at second order in the controlling coefficients. The nonbirefringent *Ansatz* of prevents birefringence to occur only at leading order in Lorentz violation. Now the classical Lagrange functions for all cases previously introduced are given as follows. The derivation for one particular of those is shown in and it works analogously for the remaining ones. For the isotropic case (denoted as $\circledcirc$) the Lagrange functions read as $$\begin{aligned} \label{eq:massive-lagrangian-isotropic} L|_{\circledcirc}^{\pm}&=\pm m_{\upgamma}\sqrt{a_{\mu\nu}u^{\mu}u^{\nu}}\,, \\[2ex] (a_{\mu\nu})&=\mathrm{diag}\left(\frac{1}{1+\kappa_{\mathrm{tr}}},-\frac{1}{1-\kappa_{\mathrm{tr}}},-\frac{1}{1-\kappa_{\mathrm{tr}}},-\frac{1}{1-\kappa_{\mathrm{tr}}}\right)=(a^{\mu\nu})^{-1}\,.\end{aligned}$$ For the nonbirefringent, anisotropic case ($\varobar$) they are given by $$\begin{aligned} \label{eq:massive-lagrangian-anisotropic} L|_{\varobar}^{\pm}&=\pm m_{\upgamma}\sqrt{b_{\mu\nu}u^{\mu}u^{\nu}}\,, \\[2ex] (b_{\mu\nu})&=\mathrm{diag}\left(\frac{1}{1+(3/2)\widetilde{\kappa}_{e-}^{11}},-\frac{1}{1+(3/2)\widetilde{\kappa}_{e-}^{11}},-\frac{1}{1+(3/2)\widetilde{\kappa}_{e-}^{11}},-\frac{1}{1-(3/2)\widetilde{\kappa}_{e-}^{11}}\right) \notag \\ &=(b^{\mu\nu})^{-1}\,,\end{aligned}$$ Finally for the first dispersion relation of the parity-odd case ($\otimes$) we obtain $$\begin{aligned} \label{eq:massive-lagrangian-parity-odd} L|_{\otimes}^{\pm}&=\pm m_{\upgamma}\sqrt{c_{\mu\nu}u^{\mu}u^{\nu}}\,, \displaybreak[0]\\[2ex] (c_{\mu\nu})&=\begin{pmatrix} 1/(1+\mathcal{E}^2) & 0 & 0 & -\mathcal{E}/(1+\mathcal{E}^2) \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -\mathcal{E}/(1+\mathcal{E}^2) & 0 & 0 & -1/(1+\mathcal{E}^2) \\ \end{pmatrix}=(c^{\mu\nu})^{-1}\,.\end{aligned}$$ Finding a classical Lagrangian that corresponds to the quartic dispersion equation of is a challenging task that we leave for the future. The examples for Lagrange functions of Lorentz-violating photons in , , and reveal the general behavior. When the photon dispersion equation is of the form $Q^{\mu\nu}k_{\mu}k_{\nu}=m_{\upgamma}^2$ with an invertible $(4\times 4)$ matrix $Q$ the associated Lagrange function generically reads as (see [@Kostelecky:2010hs] for the fermion analogue): $$L^{\pm}=\pm m_{\upgamma}\sqrt{Q^{-1}_{\mu\nu}u^{\mu}u^{\nu}}\,,$$ These Lagrange functions rely on the existence of a nonzero photon mass. In general, Lagrange functions are of mass dimension one, which is why they have to involve some dimensionful scale characteristic for the physical problem considered. For the classical fermionic point-particle analogues studied in [@Kostelecky:2010hs] this scale corresponds to the particle mass. In MCS theory the Chern-Simons mass $m_{\scriptscriptstyle{\mathrm{CS}}}$ takes the role of the characteristic dimensionful scale as we saw in . However since modified Maxwell theory does not involve a dimensionful scale, a photon mass $m_{\upgamma}$ had to be introduced to construct Lagrange functions for the classical point-particle analogues. Classical wavefront ------------------- A photon mass is undoubtedly not an attractive feature in a theory, since the mass term violates gauge invariance. Even if a photon mass has to be introduced as an intermediate ingredient to regularize infrared divergences in quantum corrections or to grant a consistent quantization of a particular Lorentz-violating framework, cf. [@Colladay:2014dua], it should be possible to consider the limit $m_{\upgamma}\mapsto 0$ at the end of any calculation. For this reason an alternative procedure shall be developed to obtain the classical analogue of (Lorentz-violating) photons. Classically, an electromagnetic pulse makes up a wavefront that can be interpreted as a surface in four-dimensional spacetime: $w=w(t,\mathbf{x})=0$. In a Lorentz-invariant theory it fulfills the following equation [@Fock:1959]: $$\left(\frac{\partial w}{\partial t}\right)^2-(\boldsymbol{\nabla}w)^2=0\,.$$ Computing the square root and choosing one particular sign results in: $$\label{eq:wave-front-equation} \frac{\partial w}{\partial t}-\sqrt{\left(\frac{\partial w}{\partial x}\right)^2+\left(\frac{\partial w}{\partial y}\right)^2+\left(\frac{\partial w}{\partial z}\right)^2}=0\,.$$ The latter is a Hamilton-Jacobi equation where $w$ is understood as the action $S$ and the expression on the right-hand side as the Hamilton function: $$\label{eq:hamilton-jacobi-equation} \frac{\partial S}{\partial t}+H(\mathbf{x},\boldsymbol{\nabla}S)=0\,,\quad S(t,\mathbf{x})=w(t,\mathbf{x})\,,\quad H(\mathbf{x},\mathbf{k})=-\sqrt{\mathbf{k}^2}\,,$$ where $\mathbf{k}$ is the wave vector (momentum). Examples that obey are $$\begin{aligned} w&=t-\widehat{\mathbf{a}}\cdot \mathbf{x}\,,\quad |\widehat{\mathbf{a}}|=1\,, \\[2ex] w&=t-\sqrt{\mathbf{x}^2}\,.\end{aligned}$$ The first describes a plane wavefront with unit normal vector $\widehat{\mathbf{a}}$ and the second a spherical wavefront. This can be seen by equating $w$ with zero and considering a fixed value for $t$. Introducing $\lambda$ as a parameter for the trajectory of the wave, both wavefronts can be differentiated with respect to $\lambda$, which leads to $$\begin{aligned} \frac{\partial w}{\partial\lambda}&=u^0-\widehat{\mathbf{a}}\cdot\mathbf{u}\,, \\[2ex] \frac{\partial w}{\partial\lambda}&=u^0-\sqrt{\mathbf{u}^2}\,,\quad u^0\equiv \frac{\mathrm{d}t}{\mathrm{d}\lambda}\,,\quad \mathbf{u}\equiv \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\lambda}\,.\end{aligned}$$ At a first glance it may be assumed that the latter are suitable Lagrange functions, since they are positively homogeneous of degree one. However computing the derived metrics $g_{\mu\nu}$ according to $$g_{\mu\nu}\equiv\frac{1}{2}\frac{\partial^2L^2}{\partial u^{\mu}\partial u^{\nu}}\,,$$ quickly reveals that their resulting determinants vanish. Therefore such a $g_{\mu\nu}$ is not invertible and definitely fails to describe a possible (pseudo)-Finsler structure. This is a result that can be shown to hold in general. Assume that a Lagrange function $L$ exists describing the classical wave-front analogue of photons. Then the associated conjugated momentum $p_{\mu}$ must be lightlike to obey the photon dispersion relation: $$\label{eq:lightlike-canonical-momentum} p_{\mu}=-\frac{\partial L}{\partial u^{\mu}}\,,\quad p_{\mu}=-f(u^0,u)\begin{pmatrix} 1 \\ \pm 1 \\ \end{pmatrix}_{\mu}\,.$$ Due to rotational symmetry in the Lorentz-invariant case it is sufficient to consider a (1+1)-dimensional spacetime, which is why a lightlike $p_{\mu}$ must be of the form stated in with a $C^{\infty}$ function $f(u^0,u)$ where $u\equiv |\mathbf{u}|$. The derived metric is then given by $$\begin{aligned} g_{\mu\nu}&=\frac{1}{2}\frac{\partial^2 L^2}{\partial u^{\mu}\partial u^{\nu}}= L\frac{\partial^2 L}{\partial u^{\mu}\partial u^{\nu}}+\frac{\partial L}{\partial u^{\mu}}\frac{\partial L}{\partial u^{\nu}} \notag \\ &= L\begin{pmatrix} f^{(1)} & f^{(2)} \\ \pm f^{(1)} & \pm f^{(2)} \\ \end{pmatrix}_{\mu\nu}+\begin{pmatrix} f^2 & \pm f^2 \\ \pm f^2 & f^2 \\ \end{pmatrix}_{\mu\nu}=\begin{pmatrix} Lf^{(1)}+f^2 & Lf^{(2)}\pm f^2 \\ \pm( Lf^{(1)}+f^2) & \pm( Lf^{(2)}\pm f^2) \\ \end{pmatrix}_{\mu\nu}\,,\end{aligned}$$ where $(1)$ denotes differentiation with respect to $u^0$ and $(2)$ means differentiation by $u$. It clearly holds that $\det(g_{\mu\nu})=0$ irrespective of the unknown function $f$. Therefore a Lagrange function $ L$ with an invertible derived metric cannot exist in the photon case. Because of this an alternative procedure has to be developed to assign a possible (pseudo)-Finsler structure to photons, which will be examined in what follows. Finsler structures of the photon sector {#sec:finsler-structures-photon} ======================================= In the previous section it was motivated that the usual method to finding Finsler structures in the fermion sector does not seem to work in the minimal [*CPT*]{}-even photon sector. The reason is the absence of a dimensionful physical scale needed for dimensional consistency of a Lagrange function. Photons must be treated differently from fermions to obtain something like a classical description. This shall be undertaken in the current section. Lorentz-invariant case {#sec:lorentz-invariant-case} ---------------------- To become familiar with our goals, the situation in standard electrodynamics will be described first. In a Lorentz-invariant vacuum Maxwell’s equations in momentum space read as follows: $$\begin{aligned} \label{eq:maxwell-equations-1} \mathbf{k}\times \mathbf{B}+\omega\mathbf{E}&=\mathbf{0}\,,\quad \mathbf{k}\times \mathbf{E}-\omega\mathbf{B}=\mathbf{0}\,, \\[2ex] \label{eq:maxwell-equations-2} \mathbf{k}\cdot \mathbf{E}&=0\,,\quad \mathbf{k}\cdot \mathbf{B}=0\,.\end{aligned}$$ Here $\mathbf{E}$ is the electric field, $\mathbf{B}$ the magnetic flux density, $\mathbf{k}$ the wave vector, and $\omega$ the frequency. The dispersion relation can be derived directly from the wave equation. The latter is obtained by computing the cross product of the wave vector and, e.g., the first of where the second equation has to be plugged in subsequently: $$\label{eq:lorentz-invariant-wave-equation} \mathbf{k}\times (\mathbf{k}\times \mathbf{B})+\omega\,\mathbf{k}\times \mathbf{E}=\mathbf{k}(\mathbf{k}\cdot \mathbf{B})-\mathbf{k}^2\mathbf{B}+\omega^2\,\mathbf{B}=(\omega^2-\mathbf{k}^2)\mathbf{B}=\mathbf{0}\,.$$ Here the second of is used as well, which says that in a Lorentz-invariant vacuum the magnetic field is transverse. Equation (\[eq:lorentz-invariant-wave-equation\]) has nontrivial solutions for the magnetic field only in case of $\omega^2=\mathbf{k}^2$, which immediately leads to the dispersion relation $\omega=|\mathbf{k}|$ of electromagnetic waves. The dispersion equation $$\label{eq:dispersion-equation-standard} \omega^2-\mathbf{k}^2=0\,,$$ is the base to determine the Finsler structure associated to standard Maxwell theory. The method is introduced in [@Antonelli:1993] and will be described as follows. Let $M$ be a Finsler manifold and $F=F(x,y)$ the corresponding Finsler structure with $x\in M$ and $y\in T_xM$ where $T_xM$ is the tangent space at $x$. The indicatrix $S_xM$ at a point $x$ of a Finsler space is the set of all $y$ where the Finsler structure takes the constant value 1, i.e., $S_xM=\{y\in T_xM|F(x,y)=1\}$. Note that a Finsler structure defines an indicatrix, but conversely each indicatrix determines a Finsler structure [@Constantinescu:2009]. Finsler himself expressed the idea that an indicatrix might model the phase velocity of light waves in both isotropic and anisotropic materials. Hence what is needed to associate a Finsler structure to a photon theory is an indicatrix [@Antonelli:1993]. The phase velocity vector is defined as $\mathbf{v}_{\mathrm{ph}}\equiv \widehat{\mathbf{k}}v_{\mathrm{ph}}$ with $v_{\mathrm{ph}}=\omega/|\mathbf{k}|$ and the unit wave vector is $\widehat{\mathbf{k}}\equiv \mathbf{k}/|\mathbf{k}|$. Since still depends on both the energy and the momentum components, we divide it by $|\mathbf{k}|^2$. This results in an equation that involves the phase velocity and quantities of zero mass dimension: $$\label{eq:phase-velocity-standard} v_{\mathrm{ph}}^2-1=0\,.$$ Now can be considered as the indicatrix of the associated Finsler structure that it still to be found. This is accomplished using Okubo’s technique, which is outlined in [@Antonelli:1993; @Bao:2000]. Consider a surface within a Finsler manifold $M$ that is described by an equation $f(x,y)=0$. A function $F(y)$ taking a constant value 1 on such a surface can be found by solving the equation $f(x,y/F(y))=0$ with respect to $F(y)$ where the solution does not necessarily have to be unique. Denoting the phase velocity by $v_{\mathrm{ph}}\equiv |\mathbf{u}|$ with $\mathbf{u}\equiv (u^1,u^2,u^3)$ we perform the replacement $u^i\mapsto u^i/F(\mathbf{u})$ and obtain from $$\frac{\mathbf{u}^2}{F(\mathbf{u})^2}-1=0\,.$$ The latter can be solved for $F(\mathbf{u})$ immediately: $$\label{eq:finsler-structure-standard} F(\mathbf{u})|_{\text{LI}}^{\pm}=\pm\sqrt{\mathbf{u}^2}=\pm\sqrt{r_{ij}u^iu^j}\,,\quad r_{ij}=\mathrm{diag}(1,1,1)_{ij}\,.$$ As long as the intrinsic metric $r_{ij}$ is positive definite, which is the case for the particular $r_{ij}$ given, $F(\mathbf{u})|_{\text{LI}}^{+}$ fulfills all properties of . Therefore it can be interpreted as a three-dimensional Finsler structure where the derived metric $g_{\mathrm{LI},ij}^{\pm}$ corresponds to the intrinsic metric. Since the Cartan torsion vanishes, it must be a Riemannian structure according to Deicke’s theorem. Isotropic case {#sec:isotropic-case} -------------- In the Lorentz-violating case modified Maxwell’s equations can be constructed by using Eqs. (4) – (6) of [@Kostelecky:2002hh]. A Lorentz-violating vacuum behaves like an effective medium for electromagnetic waves, which is why Maxwell’s equations now involve nontrivial permeability and permittivity tensors. In momentum space they read as follows (where the spatial indices of $\mathbf{k}$ are understood to be upper ones): $$\begin{aligned} \label{eq:maxwell-equations-modified-1} \mathbf{k}\times \mathbf{H}+\omega\mathbf{D}&=\mathbf{0}\,,\quad \mathbf{k}\times \mathbf{E}-\omega\mathbf{B}=\mathbf{0}\,, \\[2ex] \label{eq:maxwell-equations-modified-2} \mathbf{k}\cdot \mathbf{D}&=0\,,\quad \mathbf{k}\cdot \mathbf{B}=0\,.\end{aligned}$$ The first two of these deliver relationships between the electric displacement $\mathbf{D}$, the magnetic field $\mathbf{H}$, the electric field $\mathbf{E}$, and the magnetic flux density $\mathbf{B}$. The transformation between $(\mathbf{D},\mathbf{H})$ and $(\mathbf{E},\mathbf{B})$ is governed by $(3\times 3)$ matrices $\kappa_{\scriptscriptstyle{DE}}$, $\kappa_{\scriptscriptstyle{DB}}$, $\kappa_{\scriptscriptstyle{HE}}$, and $\kappa_{\scriptscriptstyle{HB}}$ comprising the controlling coefficients and they are given by Eq. (4) in the latter reference. In the isotropic case considered here the matrices $\kappa_{\scriptscriptstyle{DB}}$ and $\kappa_{\scriptscriptstyle{HE}}$ do not contribute. It then holds that \[eq:isotropic-case-property-tensors\] $$\begin{aligned} \mathbf{H}&=\mu^{-1}\mathbf{B}\,,\quad \mu^{-1}=\mathds{1}_3+\kappa_{\scriptscriptstyle{HB}}=\mathds{1}_3-\kappa_{\scriptscriptstyle{DE}}\,, \\[2ex] \mathbf{D}&=\varepsilon\mathbf{E}\,,\quad \varepsilon=\mathds{1}_3+\kappa_{\scriptscriptstyle{DE}}\,, \\[2ex] \kappa_{\scriptscriptstyle{DE}}&=\widetilde{\kappa}_{\mathrm{tr}}\,\mathrm{diag}(1,1,1)=-\kappa_{\scriptscriptstyle{HB}}\,, \\[2ex] \varepsilon\mu&=n^2\,\mathrm{diag}(1,1,1)\,,\quad n^{-1}=\mathcal{A}\equiv \sqrt{\frac{1-\kappa_{\mathrm{tr}}}{1+\kappa_{\mathrm{tr}}}}\,.\end{aligned}$$ Maxwell’s equations in momentum space will be needed to obtain the dispersion relations. Each of the equations involves different fields. However to obtain the dispersion relation, a single equation is required that contains one of the four fields only. Since according to the different fields are related by matrices proportional to the unit matrix, the standard procedure outlined in works here: $$\begin{aligned} \mathbf{k}\times (\mathbf{k}\times \mathbf{E})-\omega(\mathbf{k} \times \mathbf{B})&=\mathbf{k}\times (\mathbf{k}\times \mathbf{E})-\omega\mu(\mathbf{k}\times \mathbf{H}) \notag \\ &=\mathbf{k}\times (\mathbf{k}\times \mathbf{E})+\omega^2\mu\mathbf{D}=\mathbf{k}\times (\mathbf{k}\times \mathbf{E})+\omega^2\varepsilon\mu\mathbf{E}=\mathbf{0}\,.\end{aligned}$$ Writing the equation explicitly in matrix form leads to $$\begin{pmatrix} n^2\omega^2-(k_2^2+k_3^2) & k_1k_2 & k_1k_3 \\ k_1k_2 & n^2\omega^2-(k_1^2+k_3^2) & k_2k_3 \\ k_1k_3 & k_2k_3 & n^2\omega^2-(k_1^2+k_2^2) \\ \end{pmatrix}\begin{pmatrix} E^1 \\ E^2 \\ E^3 \\ \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}\,,$$ where $\mathbf{E}\equiv (E^1,E^2,E^3)^T$ is the electric field strength vector. Lowering the indices of the components of $\mathbf{k}$ does not lead to changes, since the components always appear in bilinear combinations. The condition of a vanishing determinant of the coefficient matrix, which is demanded for the existence of nontrivial solutions for the electric field, leads to the dispersion equation $$\label{eq:dispersion-relation-isotropic} 0=n^2\omega^2(n^2\omega^2-\mathbf{k}^2)^2\,.$$ From this we obtain the spurious solution $\omega=0$ associated to a nonpropagating wave and the modified dispersion relation $\omega=\mathcal{A}|\mathbf{k}|$. Now we again need an indicatrix. A reasonable choice to start with is . Dividing the latter by the prefactor and computing the square root does not change the set of physical zeros for $\omega$, i.e., we can also take $$n^2\omega^2-\mathbf{k}^2=0\,.$$ A subsequent division by $|\mathbf{k}|^2$ results in the indicatrix of the related Finsler structure: $$\label{eq:phase-velocity-isotropic} v_{\mathrm{ph}}^2-\mathcal{A}^2=0\,.$$ Using Okubo’s technique we obtain $F(\mathbf{u})$ immediately: $$\begin{aligned} 0&=\frac{\mathbf{u}^2}{F(\mathbf{u})^2}-\mathcal{A}^2\,, \\[2ex] \label{eq:finsler-structure-isotropic} F(\mathbf{u})|_{\circledcirc}^{\pm}&=\pm\frac{1}{\mathcal{A}}\sqrt{\mathbf{u}^2}=\pm\frac{1}{\mathcal{A}}\sqrt{r_{ij}u^iu^j}\,,\quad r_{ij}=\mathrm{diag}(1,1,1)_{ij}\,,\end{aligned}$$ where the symbol $\circledcirc$ denotes “isotropic.” Comparing to we see that the only difference in comparison to the Lorentz-invariant case is the prefactor $1/\mathcal{A}$. This is not surprising, as the case considered is isotropic and the result involves the spatial velocity components only. For a positive definite $r_{ij}$, $F(\mathbf{u})|_{\circledcirc}^{+}$ fulfills all properties of a Finsler structure where the derived metric is given by $g_{\circledcirc,ij}^{\pm}=r_{ij}/\mathcal{A}^2$. Due to the isotropy the latter is still Riemannian, which can be explicitly checked via the Cartan torsion. In comparison to the Lorentz-invariant case it involves a global scaling factor. Anisotropic, nonbirefringent case {#sec:anisotropic-nonbirefringent-case} --------------------------------- The anisotropic case with a single modified dispersion relation reveals some peculiar properties. The matrices relating the different electromagnetic fields with each other are given by $$\begin{aligned} \label{eq:medium-tensors-anisotropic-nonbirefringent} \kappa_{\scriptscriptstyle{DE}}&=\frac{3}{2}\widetilde{\kappa}_{e-}^{11}\,\mathrm{diag}(1,1,-1)=-\kappa_{\scriptscriptstyle{HB}}\,, \\[2ex] \kappa_{\scriptscriptstyle{DB}}&=\kappa_{\scriptscriptstyle{HE}}=\mathbf{0}_3\,,\end{aligned}$$ with the $(3\times 3)$ zero matrix $\mathbf{0}_3$. The matrices $\kappa_{\scriptscriptstyle{DE}}$ and $\kappa_{\scriptscriptstyle{HB}}$ are diagonal as well, but the difference to the isotropic case is that they are no longer proportional to the identity matrix. This is not surprising due to the preferred spacelike direction $\boldsymbol{\zeta}$ pointing along the third spatial axis where there is a residual isotropy in the plane perpendicular to this axis. Therefore the first two components of the diagonal matrix $\varepsilon\mu$ are equal, but the third differs from those: $$\varepsilon\mu=\mathrm{diag}(n_1^2,n_2^2,n_3^2)\,,\quad n_1=n_2=\frac{1}{\mathcal{B}}\,,\quad n_3=\mathcal{B}\,,\quad \mathcal{B}\equiv \sqrt{\frac{1-(3/2)\widetilde{\kappa}_{e-}^{11}}{1+(3/2)\widetilde{\kappa}_{e-}^{11}}}\,.$$ Now we again need an equation that can serve as a basis for the indicatrix of the associated Finsler space. Multiplying the second of with $\mu^{-1}$, computing the cross product with $\mathbf{k}$, and using the first of leads to an equation for the electric field vector: $$\mathbf{k}\times \left[(\mu^{-1}(\mathbf{k}\times \mathbf{E})\right]+\omega^2\varepsilon \mathbf{E}=\mathbf{0}\,.$$ Multiplying the latter with an appropriate prefactor, in matrix form it reads as follows: $$\begin{pmatrix} \omega^2-k_2^2-k_3^2n_3^2 & k_1k_2 & k_1k_3n_3^2 \\ k_1k_2 & \omega^2-k_1^2-k_3^2n_3^2 & k_2k_3n_3^2 \\ k_1k_3n_3^2 & k_2k_3n_3^2 & (\omega^2-k_1^2-k_2^2)n_3^2 \\ \end{pmatrix}\begin{pmatrix} E^1 \\ E^2 \\ E^3 \\ \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}\,.$$ Lowering the components of $\mathbf{k}$ does not produce any changes. The determinant condition for this system of equations leads to $$\begin{aligned} 0&=\omega^2(\omega^2-k_{\bot}^2-k_{\scalebox{0.6}{$\|$}}^2n_3^2)^2\,, \\[2ex] k_{\scalebox{0.6}{$\|$}}&\equiv \mathbf{k}\cdot \boldsymbol{\zeta}\,,\quad k_{\bot}\equiv |\mathbf{k}-k_{\scalebox{0.6}{$\|$}}\boldsymbol{\zeta}|\,.\end{aligned}$$ For convenience the three-momentum $\mathbf{k}$ is decomposed into a component $k_{\scalebox{0.6}{$\|$}}$ along the preferred spatial direction $\boldsymbol{\zeta}=(0,0,1)^T$ and into a component $k_{\bot}$ perpendicular to $\boldsymbol{\zeta}$. This results in the spurious solution $\omega=0$ and a single dispersion relation for electromagnetic waves: $$\label{eq:dispersion-relation-anisotropic} \omega=\sqrt{k_{\bot}^2+\mathcal{B}^2k_{\scalebox{0.6}{$\|$}}^2}\,.$$ Here the remaining isotropy perpendicular to the preferred direction becomes evident as well. The photon will only be affected by Lorentz violation in case it has a momentum component pointing along the preferred direction. Note that the result of is very interesting from the perspective that the underlying Lorentz-violating framework is anisotropic, but in spite of this anisotropy there is only a single dispersion relation. In contrast, birefringence, i.e., the property of having two different dispersion laws dependent on photon polarization seems to always occur in anisotropic media in nature. The reason that there is a single dispersion relation here only is the extreme fine tuning of permeability and permittivity (cf. ), which can most probably not be found in any materials. Now, the equation for the indicatrix follows from $$\omega^2-k_{\bot}^2-k_{\scalebox{0.6}{$\|$}}^2n_3^2=0\,,$$ in dividing it by $\mathbf{k}^2$. Introducing the angle $\vartheta$ between the wave vector $\mathbf{k}$ and the preferred spatial direction $\boldsymbol{\zeta}$ leads to $$\begin{aligned} \label{eq:pre-indicatrix-anisotropic} 0&=v_{\mathrm{ph}}^2-\sin^2\vartheta-\mathcal{B}^2\cos^2\vartheta\,, \\[2ex] \cos\vartheta&\equiv \widehat{\mathbf{k}}\cdot \boldsymbol{\zeta}\,,\quad \widehat{\mathbf{k}}\equiv \frac{\mathbf{k}}{|\mathbf{k}|}\,.\end{aligned}$$ Thinking of $\vartheta$ as the polar angle in spherical coordinates, can be reinterpreted using $$v_{\mathrm{ph}}^2=\mathbf{u}^2\,,\quad \cos\vartheta=\frac{u^3}{|\mathbf{u}|}\,,\quad \sin\vartheta=\frac{\sqrt{(u^1)^2+(u^2)^2}}{|\mathbf{u}|}\,,$$ as follows: $$\label{eq:indicatrix-anisotropic} \mathbf{u}^4-\left[(u^1)^2+(u^2)^2+\mathcal{B}^2(u^3)^2\right]=0\,.$$ The latter is the equation that determines the indicatrix. Okubo’s technique can again be used to obtain a Finsler structure directly when $\mathbf{u}$ is replaced by $\mathbf{u}/F(\mathbf{u})$ in : $$\mathbf{u}^4-F(\mathbf{u})^2\left[(u^1)^2+(u^2)^2+\mathcal{B}^2(u^3)^2\right]=0\,.$$ This leads to the result $$\label{eq:finsler-structure-anisotropic} F(\mathbf{u})|_{\varobar}^{\pm}=\pm\frac{\mathbf{u}^2}{\sqrt{(u^1)^2+(u^2)^2+\mathcal{B}^2(u^3)^2}}\,,$$ which can also be written in the form $$\label{eq:finsler-structure-anisotropic} F(\mathbf{u})|_{\varobar}^{\pm}=\pm\frac{r_{ij}u^iu^j}{\sqrt{s_{ij}u^iu^j}}\,,\quad r_{ij}=\mathrm{diag}(1,1,1)_{ij}\,,\quad s_{ij}=\mathrm{diag}(1,1,\mathcal{B}^2)_{ij}\,.$$ Here $\varobar$ means “anisotropic.” In principle the Finsler structure can be interpreted to involve an intrinsic metric $r_{ij}$ and a second metric $s_{ij}$. Since the background considered is flat, it is reasonable to take $r_{ij}$ as the metric that determines the lengths of vectors and the angles between vectors. For general $r_{ij}$ and $s_{ij}$ the derived metric is given by $$\begin{aligned} g_{ij}&=F(\mathbf{u})|_{\varobar}^{\pm}\frac{\partial^2F(\mathbf{u})|_{\varobar}^{\pm}}{\partial u^i\partial u^j}+\frac{\partial F(\mathbf{u})|_{\varobar}^{\pm}}{\partial u^i}\frac{\partial F(\mathbf{u})|^{\pm}_{\varobar}}{\partial u^j}\,, \displaybreak[0]\\[2ex] \frac{\partial F(\mathbf{u})|_{\varobar}^{\pm}}{\partial u^i}&=\pm\frac{1}{(s_{ab}u^au^b)^{3/2}}Q_{iklm}u^ku^lu^m\,, \displaybreak[0]\\[2ex] \frac{\partial^2F(\mathbf{u})|_{\varobar}^{\pm}}{\partial u^i\partial u^j}&=\mp\frac{3s_{jn}}{(s_{ab}u^au^b)^{5/2}}Q_{iklm}u^ku^lu^mu^n \notag \displaybreak[0]\\ &\phantom{{}={}}\pm\frac{1}{(s_{ab}u^au^b)^{3/2}}Q_{iklm}\left(\delta^{kj}u^lu^m+\delta^{jl}u^ku^m+\delta^{mj}u^ku^l\right)\,, \displaybreak[0]\\[2ex] Q_{iklm}&=2s_{kl}r_{im}-r_{kl}s_{im}\,.\end{aligned}$$ This result is not very illuminating. When contracted with appropriate velocity components it collapses to $(F(\mathbf{u})|_{\varobar}^{\pm})^2$, which follows from its homogeneity of degree 2: $$g^{\pm}_{\varobar,ij}u^iu^j=(F(\mathbf{u})|_{\varobar}^{\pm})^2\,,\quad g^{\pm}_{\varobar,ij}\equiv \frac{1}{2}\frac{\partial^2(F(\mathbf{u})|_{\varobar}^{\pm})^2}{\partial u^i\partial u^j}\,.$$ Now the following properties of $F(\mathbf{u})|_{\varobar}^{\pm}$ can be deduced: - $F(\mathbf{u})|_{\varobar}^+>0$ if $r_{ij}$ is positive definite, - $F(\mathbf{u})|_{\varobar}^{\pm}\in C^{\infty}$ for $\mathbf{u}\in TM\setminus \{0\}$ as well as positive definite $s_{ij}$, - $F(\lambda\mathbf{u})|_{\varobar}^{\pm}=\lambda F(\mathbf{u})|_{\varobar}^{\pm}$ for $\lambda>0$, i.e., positive homogeneity, - and the derived metric $g_{ij}$ is positive definite as long as $s_{ij}$ is positive definite. Therefore as long as both $r_{ij}$ and $s_{ij}$ are positive definite, which in particular is the case for $r_{ij}$ and $s_{ij}$ given in , $F(\mathbf{u})|_{\varobar}^+$ defines a three-dimensional Finsler structure, indeed. Furthermore both the Cartan and the Matsumoto torsion can be computed to be able to classify this Finsler structure. The results are complicated and they do not provide further insight, which is why they will be omitted. However they are nonzero in general whereby according to Deicke’s theorem, is not a Riemannian structure and according to the Matsumoto-Hōjō theorem it is neither a Randers nor a Kropina structure. The result corresponds to Eq. (4.2.2.6) of [@Antonelli:1993] where $a=\mathcal{B}$ and $b=1$ in their notation. They denote this type of Finsler structure as a second-order Kropina structure in resemblance to a Kropina structure $F(\mathbf{u})=\alpha^2/\beta$ with $\alpha=\sqrt{a_{ij}u^iu^j}$ and $\beta=b_iu^i$. In the latter reference appears in the context of light propagation in uniaxial media. The numerator involves the Euclidean intrinsic metric $r_{ij}$ only, whereas the denominator is characterized by another metric $s_{ij}$. The latter could be thought of as the metric governing physics, since it involves the physical quantity $\mathcal{B}$. Anisotropic, birefringent (at second order) case {#sec:anisotropic-birefringent-case} ------------------------------------------------ The penultimate example provides a case of modified Maxwell theory that has not been considered in . It is parity-even and characterized by two preferred spacelike directions:$$\label{eq:preferred-directions-anisotropic-case} (\zeta_1^{\mu})=\begin{pmatrix} 0 \\ \boldsymbol{\zeta}_1 \\ \end{pmatrix}\,,\quad (\zeta_2^{\mu})=\begin{pmatrix} 0 \\ \boldsymbol{\zeta}_2 \\ \end{pmatrix}\,,\quad \boldsymbol{\zeta}_1=\begin{pmatrix} \sin\eta \\ 0 \\ \cos\eta \\ \end{pmatrix}\,,\quad \boldsymbol{\zeta}_2=\begin{pmatrix} -\sin\eta \\ 0 \\ \cos\eta \\ \end{pmatrix}\,.$$ They are normalized and enclose an angle of $2\eta$. We consider an observer frame with one nonzero controlling coefficient $\mathcal{G}$. Then the $(4\times 4)$ matrix employed in the nonbirefringent *Ansatz* reads $$\widetilde{\kappa}^{\mu\nu}=\mathcal{G}\left(\zeta_1^{\mu}\zeta_2^{\nu}+\zeta_1^{\nu}\zeta_2^{\mu}-\frac{1}{2}(\zeta_1\cdot \zeta_2)\eta^{\mu\nu}\right)\,.$$ This corresponds to the following choices for the matrices that appear in Maxwell’s equations: $$\begin{aligned} \kappa_{\scriptscriptstyle{DE}}&=\mathcal{G}\,\mathrm{diag}(1,\cos(2\eta),-1)=-\kappa_{\scriptscriptstyle{HB}}\,, \\[2ex] \kappa_{\scriptscriptstyle{DB}}&=\kappa_{\scriptscriptstyle{HE}}=\mathbf{0}_3\,.\end{aligned}$$ Hence there are nontrivial permeability and permittivity tensors, but the electric and magnetic fields do still not mix. Using these matrices, modified Maxwell’s equations can be obtained according to the procedure used in . The condition of a vanishing coefficient determinant for nontrivial solutions results in an equation for the dispersion relation: $$\begin{aligned} \label{eq:dispersion-relation-off-shell-anisotropic-nonbirefringent} 0&=\omega^2\left[(1-\mathcal{G}^2)\omega^2-(1+\mathcal{G})[1-\mathcal{G}\cos(2\eta)]k_1^2-(1-\mathcal{G}^2)k_2^2-(1-\mathcal{G})[1-\mathcal{G}\cos(2\eta)]k_3^2\right] \notag \\ &\phantom{{}={}}\times \left\{\omega^2[1+\mathcal{G}\cos(2\eta)]-(1+\mathcal{G})k_1^2-[1+\mathcal{G}\cos(2\eta)]k_2^2-(1-\mathcal{G})k_3^2\right\}\,.\end{aligned}$$ In contrast to the anisotropic case considered in the current framework is characterized by two distinct modified dispersion relations. They can be written in the form \[eq:dispersion-relations-anisotropic-birefringent\] $$\begin{aligned} \omega_1&=\sqrt{\mathcal{G}_1k_1^2+k_2^2+\mathcal{G}_2k_3^2}\,, \\[2ex] \omega_2&=\sqrt{\widetilde{\mathcal{G}}_1k_1^2+k_2^2+\widetilde{\mathcal{G}}_2k_3^2}\,, \\[2ex] \label{eq:finsler-structure-anisotropic-birefringence-1-constants} \mathcal{G}_1&\equiv\frac{1-\mathcal{G}\cos(2\eta)}{1-\mathcal{G}}\,,\quad \mathcal{G}_2\equiv\frac{1-\mathcal{G}\cos(2\eta)}{1+\mathcal{G}}\,, \\[2ex] \label{eq:finsler-structure-anisotropic-birefringence-2-constants} \widetilde{\mathcal{G}}_1&\equiv\frac{1+\mathcal{G}}{1+\mathcal{G}\cos(2\eta)}\,,\quad \widetilde{\mathcal{G}}_2\equiv\frac{1-\mathcal{G}}{1+\mathcal{G}\cos(2\eta)}\,.\end{aligned}$$ Evidently the contribution associated to the second three-momentum component stays unmodified which is reasonable, since the preferred directions of do not point along the second spatial axis. Each dispersion relation can be expanded for $\mathcal{G}\ll 1$ showing that they differ at second order in Lorentz violation. In general the nonbirefringent *Ansatz* of works at leading order only. Besides, the dispersion relations depend on the angle $2\eta$ enclosed by the two preferred directions. With the normalized propagation direction of the electromagnetic wave given by $\widehat{\mathbf{k}}$, the latter encloses the angles $\theta_1$, $\theta_2$ with the first and the second preferred direction, respectively. These are given by: $$\begin{aligned} \cos\theta_1&=\widehat{\mathbf{k}}\cdot \boldsymbol{\zeta}_1=\widehat{k}^1\sin\eta+\widehat{k}^3\cos\eta\,, \\[2ex] \cos\theta_2&=\widehat{\mathbf{k}}\cdot \boldsymbol{\zeta}_2=-\widehat{k}^1\sin\eta+\widehat{k}^3\cos\eta\,.\end{aligned}$$ The components of the propagation direction vector $\widehat{\mathbf{k}}$ can now be expressed in terms of the angles $\theta_1$, $\theta_2$, and $\eta$. Note that $\widehat{\mathbf{k}}$ is a unit vector by construction: $$\label{eq:direction-angle} \widehat{k}^1=\frac{\cos\theta_1-\cos\theta_2}{2\sin\eta}\,,\quad \widehat{k}^3=\frac{\cos\theta_1+\cos\theta_2}{2\cos\eta}\,,\quad \widehat{k}^2=\sqrt{1-(\widehat{k}^1)^2-(\widehat{k}^3)^2}\,.$$ Now the two individual factors of are considered giving the modified dispersion relations. Dividing each by the wave-vector magnitude $|\mathbf{k}|$, introducing the phase velocity, and expressing all propagation direction components by the angles of , equations for the phase velocities are obtained as before: $$\begin{aligned} \label{eq:phase-velocity-equations-anisotropic} 0&=(1-\mathcal{G}^2)v_{\mathrm{ph}}^2+\frac{\mathcal{G}}{2}\left\{4\cos(\theta_1)\cos(\theta_2)-\mathcal{G}\left[\cos(2\theta_1)+\cos(2\theta_2)\right]\right\}-1\,, \\[2ex] 0&=\left[1+\mathcal{G}\cos(2\eta)\right](1-v_{\mathrm{ph}}^2)-2\mathcal{G}\cos(\theta_1)\cos(\theta_2)\,.\end{aligned}$$ In dividing the second equation by $-[1+\mathcal{G}\cos(2\eta)]$ and expanding both equations to linear order in $\mathcal{G}$ these results correspond to each other as expected. Now we are in a position to interpret the latter equations geometrically, which will lead us directly to the Finsler structures associated to this particular sector. In doing so, the velocity $\mathbf{u}$ is introduced and both the phase velocity and the angles $\theta_1$, $\theta_2$ are expressed by the magnitude or components of $\mathbf{u}$ as follows: \[eq:correspondence-angle-velocities\] $$\begin{aligned} v_{\mathrm{ph}}&=|\mathbf{u}|\,, \\[2ex] \cos\theta_1&=\frac{u^1}{|\mathbf{u}|}\sin\eta+\frac{u^3}{|\mathbf{u}|}\cos\eta\,, \\[2ex] \cos\theta_2&=-\frac{u^1}{|\mathbf{u}|}\sin\eta+\frac{u^3}{|\mathbf{u}|}\cos\eta\,.\end{aligned}$$ Inserting those into and using Okubo’s technique leads to two distinct Finsler structures. The first is given by $$\label{eq:finsler-structure-anisotropic-birefringence-1} F(\mathbf{u})|_{\varovee}^{(1)\pm}=\pm\frac{r_{ij}u^iu^j}{\sqrt{s_{ij}u^iu^j}}\,,\quad r_{ij}=\mathrm{diag}(1,1,1)_{ij}\,,\quad s_{ij}=\mathrm{diag}(\mathcal{G}_1,1,\mathcal{G}_2)_{ij}\,,$$ and the second reads as $$\label{eq:finsler-structure-anisotropic-birefringence-2} F(\mathbf{u})|_{\varovee}^{(2)\pm}=\pm\frac{r_{ij}u^iu^j}{\sqrt{s_{ij}u^iu^j}}\,,\quad r_{ij}=\mathrm{diag}(1,1,1)_{ij}\,,\quad s_{ij}=\mathrm{diag}(\widetilde{\mathcal{G}}_1,1,\widetilde{\mathcal{G}}_2)_{ij}\,.$$ Here $\varovee$ means “anisotropic and birefringent (at second order).” The four Finsler structures obtained have a form analogous to the Finsler structure found in of . This is not surprising, since both sectors are anisotropic but parity-even. Having birefringence at second order in Lorentz violation does obviously not affect the form of the Finsler structure. In such a case we can obtain several distinct Finsler structures that differ from each other at second order in the controlling coefficients via the metrics $s_{ij}$. In the latter $s_{ij}$ differs from the standard Euclidean metric only by the component $s_{33}$. Here both $s_{11}$ and $s_{33}$ are modified by Lorentz violation where they also depend on the angle $\eta$ enclosed by the two preferred directions. The component $s_{22}$ is standard, which again reflects the fact that the preferred directions have a vanishing second component. Since $s_{ij}$ involves the physical (dimensionless) constants $\mathcal{G}_i$ and $\widetilde{\mathcal{G}}_i$ for $i=1\dots 2$, it is reasonable to say that $s_{ij}$ seems to govern the physical properties of photon propagation in these cases. Parity-odd case {#sec:parity-odd-case} --------------- The final interesting sector considered involves the three parity-odd coefficients $\widetilde{\kappa}_{o+}^{12}$, $\widetilde{\kappa}_{o+}^{31}$, and $\widetilde{\kappa}_{o+}^{23}$ and it will turn out to be the most complicated one. The preferred spacetime directions are given in and the matrices relating the electromagnetic fields to each other read $$\begin{aligned} \kappa_{\scriptscriptstyle{DE}}&=\mathbf{0}_3\,,\quad \kappa_{\scriptscriptstyle{HB}}=\mathbf{0}_3\,, \\[2ex] \kappa_{\scriptscriptstyle{DB}}&=\begin{pmatrix} 0 & \widetilde{\kappa}_{o+}^{12} & -\widetilde{\kappa}_{o+}^{31} \\ -\widetilde{\kappa}_{o+}^{12} & 0 & \widetilde{\kappa}_{o+}^{23} \\ \widetilde{\kappa}_{o+}^{31} & -\widetilde{\kappa}_{o+}^{23} & 0 \\ \end{pmatrix}\,,\quad \kappa_{\scriptscriptstyle{HE}}=-\kappa_{\scriptscriptstyle{DB}}^T=\kappa_{\scriptscriptstyle{DB}}\,.\end{aligned}$$ The relationships between the fields are given by $$\begin{aligned} \mathbf{D}&=\mathbf{E}+\kappa_{\scriptscriptstyle{DB}}\mathbf{B}\,, \\[2ex] \mathbf{H}&=\kappa_{\scriptscriptstyle{HE}}\mathbf{E}+\mathbf{B}=\kappa_{\scriptscriptstyle{DB}}\mathbf{E}+\mathbf{B}\,.\end{aligned}$$ In contrast to the aforementioned cases the parity-odd case has the peculiarity that the electric fields mix with the magnetic fields. Therefore obtaining an equation for the electric field from Maxwell’s equations is more involved here. Nevertheless it can be accomplished along the following chain of steps: $$\begin{aligned} \mathbf{0}&=\kappa_{\scriptscriptstyle{DB}}(\mathbf{k}\times \mathbf{E})-\omega\kappa_{\scriptscriptstyle{DB}}\mathbf{B}\,, \\[2ex] \mathbf{0}&=\kappa_{\scriptscriptstyle{DB}}(\mathbf{k}\times \mathbf{E})-\omega(\mathbf{D}-\mathbf{E})\,, \\[2ex] \mathbf{0}&=\kappa_{\scriptscriptstyle{DB}}(\mathbf{k}\times \mathbf{E})+\mathbf{k}\times \mathbf{H}+\omega\mathbf{E}\,, \\[2ex] \mathbf{0}&=\kappa_{\scriptscriptstyle{DB}}(\mathbf{k}\times \mathbf{E})+\mathbf{k}\times (\kappa_{\scriptscriptstyle{DB}}\mathbf{E}+\mathbf{B})+\omega\mathbf{E}\,, \\[2ex] \mathbf{0}&=\kappa_{\scriptscriptstyle{DB}}(\mathbf{k}\times \mathbf{E})+\mathbf{k}\times (\kappa_{\scriptscriptstyle{DB}}\mathbf{E})+\frac{1}{\omega}\mathbf{k}\times (\mathbf{k}\times \mathbf{E})+\omega\mathbf{E}\,.\end{aligned}$$ Inserting the explicit vectors and a subsequent multiplication with $\omega$ leads to the following system in matrix form: \[eq:modified-maxwell-equations-parity-odd\] $$\begin{aligned} \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}&=(A+B)\begin{pmatrix} E^1 \\ E^2 \\ E^3 \\ \end{pmatrix}\,, \displaybreak[0]\\[2ex] A&=\begin{pmatrix} \omega^2-(k_2^2+k_3^2) & k_1k_2 & k_1k_3 \\ k_1k_2 & \omega^2-(k_1^2+k_3^2) & k_2k_3 \\ k_1k_3 & k_2k_3 & \omega^2-(k_1^2+k_2^2) \\ \end{pmatrix}\,, \displaybreak[0]\\[2ex] B&=\omega\begin{pmatrix} -2(\widetilde{\kappa}_{o+}^{31}k_2+\widetilde{\kappa}_{o+}^{12}k_3) & \widetilde{\kappa}_{o+}^{31}k_1+\widetilde{\kappa}_{o+}^{23}k_2 & \widetilde{\kappa}_{o+}^{12}k_1+\widetilde{\kappa}_{o+}^{23}k_3 \\ \widetilde{\kappa}_{o+}^{31}k_1+\widetilde{\kappa}_{o+}^{23}k_2 & -2(\widetilde{\kappa}_{o+}^{23}k_1+\widetilde{\kappa}_{o+}^{12}k_3) & \widetilde{\kappa}_{o+}^{12}k_2+\widetilde{\kappa}_{o+}^{31}k_3 \\ \widetilde{\kappa}_{o+}^{12}k_1+\widetilde{\kappa}_{o+}^{23}k_3 & \widetilde{\kappa}_{o+}^{12}k_2+\widetilde{\kappa}_{o+}^{31}k_3 & -2(\widetilde{\kappa}_{o+}^{23}k_1+\widetilde{\kappa}_{o+}^{31}k_2) \\ \end{pmatrix}\,.\end{aligned}$$ The total system can be completely decomposed into the standard part of $A$ and a Lorentz-violating contribution comprised in $B$. Note that here $B$ gets a global minus sign when lowering the indices of the $\mathbf{k}$ components. Therefore the determinant condition results in the following equation for the photon energy: $$\label{eq:dispersion-relation-parity-odd} \omega^2(\omega^2-2\omega\,\boldsymbol{\zeta}\cdot \mathbf{k}-\mathbf{k}^2)\left[(\omega-\boldsymbol{\zeta}\cdot\mathbf{k})^2-(1+\boldsymbol{\zeta}^2)\mathbf{k}^2\right]=0\,.$$ Here $\boldsymbol{\zeta}\equiv (\widetilde{\kappa}_{o+}^{23},\widetilde{\kappa}_{o+}^{31},\widetilde{\kappa}_{o+}^{12})^T$ is the spatial part of the second preferred spacetime direction and $\mathbf{k}$ is understood to have lower components. The second and the third of the three factors can be solved for the energy giving two distinct dispersion relations: $$\begin{aligned} \label{eq:dispersion-relation-parity-odd-1} \omega_1&=\boldsymbol{\zeta}\cdot\mathbf{k}+\sqrt{\mathbf{k}^2+(\boldsymbol{\zeta}\cdot\mathbf{k})^2}\,, \\[2ex] \label{eq:dispersion-relation-parity-odd-2} \omega_2&=\boldsymbol{\zeta}\cdot\mathbf{k}+\sqrt{1+\boldsymbol{\zeta}^2}|\mathbf{k}|\,, \\[2ex] \cos\vartheta&=\widehat{\boldsymbol{\zeta}}\cdot \widehat{\mathbf{k}}\,,\quad \widehat{\boldsymbol{\zeta}}\equiv \frac{\boldsymbol{\zeta}}{\mathcal{E}}\,,\quad \widehat{\mathbf{k}}\equiv \frac{\mathbf{k}}{|\mathbf{k}|}\,,\quad \mathcal{E}\equiv |\boldsymbol{\zeta}|=\sqrt{(\widetilde{\kappa}_{o+}^{23})^2+(\widetilde{\kappa}_{o+}^{31})^2+(\widetilde{\kappa}_{o+}^{12})^2}\,.\end{aligned}$$ For convenience it is again reasonable to set up the coordinate system such that $\boldsymbol{\zeta}$ points along its third axis where $\vartheta$ is the angle between the wave vector $\mathbf{k}$ and the spatial direction. Dividing the first factor of by $\mathbf{k}^2$ then leads to $$v_{\mathrm{ph}}^2-2\mathcal{E}v_{\mathrm{ph}}\cos\vartheta-1=0\,.$$ Introducing spherical polar coordinates with $v_{\mathrm{ph}}=|\mathbf{u}|$ results in $$\mathbf{u}^2-2\mathcal{E}u^3-1=0\,.$$ This is the indicatrix for the first Finsler space that can be associated to the parity-odd case. We can employ Okubo’s technique to obtain $$\begin{aligned} 0&=\mathbf{u}^2-2\mathcal{E}F(\mathbf{u})u^3-F(\mathbf{u})^2\,, \\[2ex] \label{eq:finsler-structure-parity-odd-1a} F(\mathbf{u})|_{\otimes}^{(1)\pm}&=-\mathcal{E}u^3\pm \sqrt{\mathbf{u}^2+(\mathcal{E}u^3)^2}\,, \\[2ex] \label{eq:finsler-structure-parity-odd-1b} F(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(1)\pm}&=-\boldsymbol{\zeta}\cdot\mathbf{u}\pm \sqrt{\mathbf{u}^2+(\boldsymbol{\zeta}\cdot\mathbf{u})^2}\,.\end{aligned}$$ where is the generalization of for $\boldsymbol{\zeta}$ pointing along an arbitrary direction and the symbol $\otimes$ denotes “parity-odd.” Without loss of generality the properties of the Finsler structure can be investigated with $\boldsymbol{\zeta}$ pointing along the third axis of the coordinate system, which simplifies the calculations. The derived metric is again lengthy and does not seem to provide any deeper understanding. The derived metric contracted with the spatial velocity components leads to the square of : $$g^{(1)\pm}_{\otimes,ij}u^iu^j=(F(\mathbf{u})|_{\otimes}^{(1)\pm})^2\,,\quad g^{(1)\pm}_{\otimes,ij}\equiv \frac{1}{2}\frac{\partial^2(F(\mathbf{u})|_{\otimes}^{(1)\pm})^2}{\partial u^i\partial u^j}\,.$$ Therefore the following properties of $F(\mathbf{u})|_{\otimes}^{(1)\pm}$ in can be deduced: - $F(\mathbf{u})|_{\otimes}^{(1)+}>0$ for $\mathbf{u}\in TM\setminus \{0\}$, - $F(\mathbf{u})|_{\otimes}^{(1)\pm}\in C^{\infty}$ for $\mathbf{u}\in TM\setminus \{0\}$, - $F(\lambda\mathbf{u})|_{\otimes}^{(1)\pm}=\lambda F(\mathbf{u})|_{\otimes}^{(1)\pm}$ for $\lambda>0$, and - the derived metric of $F(\lambda\mathbf{u})|_{\otimes}^{(1)\pm}$ is positive definite for $\mathbf{u}\in TM\setminus \{0\}$. Due to the first item, only $F(\mathbf{u})|_{\otimes}^{(1)+}$ is a Finsler structure. Its Matsumoto torsion vanishes, whereas the Cartan torsion does not. Furthermore when taking into account its form, $F(\mathbf{u})|_{\otimes}^{(1)+}$ must be a Randers structure. This particular type of geometry was introduced by Randers to account for the fact that particles always move on timelike trajectories pointing forwards in time [@Randers:1941]. In contrast to General Relativity his framework incorporates an additional four-vector into the metric. However this four-vector should not be considered as a preferred spacetime direction, since it can be changed by a kind of gauge transformation without affecting the arc length travelled by a particle. In the Lorentz-violating case considered here $\boldsymbol{\zeta}$ is a preferred direction, indeed. The parity-odd framework is characterized by both a preferred timelike and a spacelike direction, cf. . For the isotropic and anisotropic cases, which are parity-even, the corresponding Finsler structures are expected to involve only bilinear expressions such as $a_{ij}u^iu^j$, since these are invariant under $u^i\mapsto u'^i=-u^i$. Due to parity violation the Finsler structure of the parity-odd case is expected to involve terms such as $b_iu^i$, though. The Randers structure is a very natural possibility with this property, but it is not the only one as we shall see below. The Finsler structure of has the same form as the corresponding dispersion relation of not taking into account additional minus signs. Such structures could be called “automorphic.” They seem to appear when the dispersion equation (here ) involves one additional parity-odd contribution. The parity-odd case of modified Maxwell theory has a second indicatrix, which follows from the second factor of using the same procedure: $$\begin{aligned} v_{\mathrm{ph}}^2-2\mathcal{E}v_{\mathrm{ph}}\cos\vartheta+\mathcal{E}^2\cos^2\vartheta-(1+\mathcal{E}^2)&=0\,, \\[2ex] \mathbf{u}^2-2\mathcal{E}u^3+\frac{\mathcal{E}^2(u^3)^2}{\mathbf{u}^2}-(1+\mathcal{E}^2)&=0\,.\end{aligned}$$ Okubo’s technique leads to $$\begin{aligned} \label{eq:finsler-structure-parity-odd-2a} F(\mathbf{u})|_{\otimes}^{(2)\pm}&=\frac{-\mathcal{E}u^3\pm \sqrt{1+\mathcal{E}^2}|\mathbf{u}|}{1+\mathcal{E}^2-(\mathcal{E}u^3)^2/\mathbf{u}^2}\,, \\[2ex] \label{eq:finsler-structure-parity-odd-2b} F(\mathbf{u})|_{\otimes}^{(2)\boldsymbol{\zeta}\pm}&=\frac{-\boldsymbol{\zeta}\cdot\mathbf{u}\pm \sqrt{1+\mathcal{E}^2}|\mathbf{u}|}{1+\mathcal{E}^2-(\boldsymbol{\zeta}\cdot\mathbf{u})^2/\mathbf{u}^2}\,.\end{aligned}$$ Let us investigate the characteristics of . We again obtain $$g^{(2)\pm}_{\otimes,ij}u^iu^j=(F(\mathbf{u})|_{\otimes}^{(2)\pm})^2\,,\quad g^{(2)\pm}_{\otimes,ij}\equiv \frac{1}{2}\frac{\partial^2(F(\mathbf{u})|_{\otimes}^{(2)\pm})^2}{\partial u^i\partial u^j}\,.$$ Hence for $F(\mathbf{u})|_{\otimes}^{(2)+}$ analogue properties hold such as for , which makes it to a Finsler structure. Note that the latter is not automorphic, since its off-shell dispersion relation in does not exclusively involve additional parity-odd terms, but also contributions like $(\boldsymbol{\zeta}\cdot \mathbf{k})^2$. For this structure the Matsumoto torsion does not vanish, which is why it is neither a Randers nor a Kropina structure. The deviation from a Randers structure is of second order in the controlling coefficients: $$F(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(2)\pm}=-\boldsymbol{\zeta}\cdot\mathbf{u}\pm \sqrt{1+\mathcal{E}^2}|\mathbf{u}|+\mathcal{O}(\widetilde{\kappa}_{o+}^2)\,.$$ Recall that the massive-photon dispersion equation of this mode, , was not quadratic, but quartic. For this reason it was challenging to derive a classical point-particle Lagrange function corresponding to the second photon polarization. It is also interesting to note that a large number of complications arise in the quantum field theory based on the parity-odd framework due to the behavior of this mode [@Schreck:2011ai]. On the contrary the first mode is much easier to handle. The Finsler structures obtained seem to reflect these properties. The first, given by , is a well-understood Randers structure, whereas the second deviates from such a structure at second order in Lorentz violation, which makes its properties much more involved to analyze. The studies carried out in the current section will prove to be useful when describing photons in the geometric-optics approximation. Thereby the eikonal equation will play an important role. How all these concepts are linked to each other will be clarified in the forthcoming part of the article. Classical ray equations {#sec:classical-equations-of-motion} ======================= Propagating electromagnetic waves can be treated in the geometric-optics approximation as long as their wave lengths can be neglected in comparison to other physical length scales. For example this is possible for waves with low energies propagating over large distances when physical phenomena related to the wave character (such as diffraction) do not play a role. This physical regime could be called “classical” and the wave then corresponds to a geometric ray. The goal of the current section is to establish *ray equations* that describe the physical behavior of propagating rays. Each electromagnetic pulse has a wavefront, which separates the region with nonzero electromagnetic fields from the region with vanishing fields. At any instant of time the wavefront can be considered as a two-dimensional surface in three-dimensional space, i.e., it can be described by an equation of the form $\psi(\mathbf{x})=t$ where $\mathbf{x}$ are spatial coordinates and $t$ is the time. The gradient $\boldsymbol{\nabla}\psi$ points along the propagation direction and it is perpendicular to the surface. There is a relation between $\boldsymbol{\nabla}\psi$ and the refractive index $n$ of the medium; it reads as $|\boldsymbol{\nabla}\psi|=n$. The latter is called the *eikonal equation* in a subset of the literature. In what follows, $n$ is assumed to depend on the position $\mathbf{x}$ only, but not on the velocity $\mathbf{u}$, i.e., $n=n(\mathbf{x})$. Consider a wave propagating along a trajectory $\mathbf{x}(s)$ where $s$ is the arc length of the curve. In this parameterization the tangent vector has magnitude 1, which is why the ray equations read as follows: $$\label{eq:equations-of-motion-light} \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}=\frac{\boldsymbol{\nabla}\psi}{|\boldsymbol{\nabla}\psi|}\,,\quad n\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}=\boldsymbol{\nabla}\psi\,.$$ Computing an additional derivative of the latter with respect to $s$, its right-hand side can be expressed in terms of the refractive index as well: $$\frac{\mathrm{d}}{\mathrm{d}s}\boldsymbol{\nabla}\psi=\left(\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\cdot \boldsymbol{\nabla}\right)\boldsymbol{\nabla}\psi=\frac{1}{n}\boldsymbol{\nabla}\psi \cdot [\boldsymbol{\nabla}(\boldsymbol{\nabla}\psi)]=\frac{1}{2n}\boldsymbol{\nabla}(\boldsymbol{\nabla}\psi)^2=\frac{1}{2n}\boldsymbol{\nabla}n^2=\boldsymbol{\nabla}n\,.$$ Trajectories may not necessarily be parameterized by arc length. For an arbitrary parameterization with parameter $t$ we obtain $$\frac{\mathrm{d}}{\mathrm{d}s}=\frac{\mathrm{d}t}{\mathrm{d}s}\frac{\mathrm{d}}{\mathrm{d}t}=\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{-1}\frac{\mathrm{d}}{\mathrm{d}t}=\frac{1}{|\mathbf{u}|}\frac{\mathrm{d}}{\mathrm{d}t}\,.$$ Now the ray equations (\[eq:equations-of-motion-light\]) can be cast into the following final form: $$\begin{aligned} \label{eq:eikonal-equation-1} \frac{\mathrm{d}}{\mathrm{d}s}\left(n\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\right)&=\boldsymbol{\nabla}n\,, \\[2ex] \label{eq:equations-of-motion-light-final} \frac{\mathrm{d}}{\mathrm{d}t}(\boldsymbol{\nabla}\psi)&=|\mathbf{u}|\boldsymbol{\nabla}n\,,\quad \boldsymbol{\nabla}\psi=n\frac{\mathbf{u}}{|\mathbf{u}|}\,.\end{aligned}$$ The literature seems to be discordant about which equation should actually be called the eikonal equation. Some sources call the first one of the eikonal equation, whereas others denote it as the vector magnitude of the second one. Note that the latter leads us back to $|\boldsymbol{\nabla}\psi|=n$ (cf. the beginning of this section). In the current paper whenever referring to the eikonal equation, we will be talking about the first one of . For clarity, the vector magnitude of the second one will be called the *wavefront equation*. Equation (\[eq:equations-of-motion-light-final\]) can be understood as the Euler-Lagrange equations resulting from the condition that the following functional becomes stationary: $$\label{eq:action-photon-trajectory} L[\mathbf{x},\mathbf{u}]=\int_A^B \mathrm{d}s\,n(\mathbf{x})=\int_{T_A}^{T_B} \mathrm{d}t\,V\,,\quad V=V(\mathbf{x},\mathbf{u})=n(\mathbf{x})|\mathbf{u}|\,.$$ The integrand of this functional is the infinitesimal optical path length and the functional itself gives the total optical path length travelled by a ray along its trajectory between two points $A$ and $B$. Here $T_A$ is the departure time of the ray at $A$ and $T_B$ the arrival time at $B$. The optical path length is defined to be the path length equivalent that light has to travel in vacuum to take the same time as for a given path in a medium with refractive index $n(\mathbf{x})$. The quantity $V$ could be interpreted as the corresponding “optical velocity.” The functional of can be understood as the base of the Fermat principle, cf. [@Perlick:2005hz; @Torrome:2012kt]. Wavefront and eikonal equation in modified Maxwell theory --------------------------------------------------------- The analogue of the wavefront equation in in the context of modified Maxwell theory was partially studied in [@Xiao:2010yx]. The authors of the latter reference chose the coefficients contained in $\kappa_{\scriptscriptstyle{DE}}$ and $\kappa_{\scriptscriptstyle{HB}}$ as nonvanishing where both the trace of these matrices and the matrices mixing electric and magnetic fields were assumed to be zero. The trace components can be restored without any effort by just replacing their $\beta_E$ by $\kappa_{\scriptscriptstyle{DE}}$ and their $\beta_B$ by $\kappa_{\scriptscriptstyle{HB}}$. The wavefront equation then follows from the matrix $M_e$ in their Eq. (38): $$M_e^{ij}=\left(1-|\boldsymbol{\nabla}\psi|^2\right)\delta^{ij}+\partial^i\psi\partial^j\psi+\kappa_{\scriptscriptstyle{DE}}^{ij}-\kappa_{\scriptscriptstyle{HB}}^{kl}\varepsilon^{ink}\varepsilon^{jml}\partial^n\psi\partial^m\psi\,,$$ where $\varepsilon^{ijk}$ is the totally antisymmetric Levi-Civita symbol in three dimensions with $\varepsilon^{123}=1$. This matrix is multiplied with the time derivatives of the fields, which are singular on the wavefront. Therefore their Eq. (33) can only have nontrivial solutions if the determinant of $M_e$ vanishes. This condition directly leads to the wavefront equation within the framework considered. For the isotropic case (cf. ), the anisotropic, nonbirefringent case (cf. ), and the anisotropic, birefringent sector (cf. ) we obtain \[eq:eikonal-equations-isotropic-and-anisotropic\] $$\begin{aligned} \label{eq:eikonal-equations-isotropic} 1&=\mathcal{A}^2|\boldsymbol{\nabla}\psi|^2\,, \displaybreak[0]\\[2ex] \label{eq:eikonal-equations-anisotropic} 1&=(\partial^1\psi)^2+(\partial^2\psi)^2+\mathcal{B}^2(\partial^3\psi)^2\,, \displaybreak[0]\\[2ex] \label{eq:eikonal-equations-anisotropic-birefringent-2} 1-\mathcal{G}^2&=|\boldsymbol{\nabla}\psi|^2+\mathcal{G}\left\{(\partial^1\psi)^2[1-\cos(2\eta)]-(\partial^3\psi)^2[1+\cos(2\eta)]\right\} \notag \\ &\phantom{{}={}}-\mathcal{G}^2\left\{(\partial^2\psi)^2+[(\partial^1\psi)^2-(\partial^3\psi)^2]\cos(2\eta)\right\}\,, \displaybreak[0]\\[2ex] \label{eq:eikonal-equations-anisotropic-birefringent-1} 1+\mathcal{G}\cos(2\eta)&=|\boldsymbol{\nabla}\psi|^2+\mathcal{G}\left\{(\partial^1\psi)^2+(\partial^2\psi)^2\cos(2\eta)-(\partial^3\psi)^2\right\}\,.\end{aligned}$$ These are the analogues of the wavefront equation $|\boldsymbol{\nabla}\psi|^2=n^2$ in modified Maxwell theory. Following the lines in connection to classical Hamilton functions can be obtained as parts of the Hamilton-Jacobi equation describing a classical ray. For the sectors considered few lines above they read as follows: $$\begin{aligned} H|_{\circledcirc}&=-\mathcal{A}\sqrt{\mathbf{k}^2}\,, \displaybreak[0]\\[2ex] H|_{\varobar}&=-\sqrt{k_1^2+k_2^2+\mathcal{B}^2k_3^2}\,, \displaybreak[0]\\[2ex] \sqrt{1-\mathcal{G}^2}H|_{\varovee}^{(1)}&=-\Big\{\mathbf{k}^2+\mathcal{G}\left\{k_1^2[1-\cos(2\eta)]-k_3^2[1+\cos(2\eta)]\right\}\Big. \notag \\ &\phantom{{}={}-\Big(}\Big.-\mathcal{G}^2\left[k_2^2+(k_1^2-k_3^2)\cos(2\eta)\right]\!\Big\}^{1/2}\,, \notag \\ H|_{\varovee}^{(1)}&=-\sqrt{\mathcal{G}_1k_1^2+k_2^2+\mathcal{G}_2k_3^2}\,, \displaybreak[0]\\[2ex] \sqrt{1+\mathcal{G}\cos(2\eta)}H|_{\varovee}^{(2)}&=-\sqrt{\mathbf{k}^2+\mathcal{G}\left[k_1^2+k_2^2\cos(2\eta)-k_3^2\right]}\,, \notag \\ H|_{\varovee}^{(2)}&=-\sqrt{\widetilde{\mathcal{G}}_1k_1^2+k_2^2+\widetilde{\mathcal{G}}_2k_3^2}\,,\end{aligned}$$ with $\mathcal{G}_1$, $\mathcal{G}_2$ of and $\widetilde{\mathcal{G}}_1$, $\widetilde{\mathcal{G}}_2$ taken from . These Hamilton functions are directly linked to the modified dispersion relations, cf. the paragraph below for the isotropic case, for the anisotropic (nonbirefringent) sector, and for the anisotropic (birefringent) case. This nicely demonstrates that all computations are consistent with each other. The wavefront equations (\[eq:eikonal-equations-isotropic-and-anisotropic\]) are not suitable for our calculations, since they involve first derivatives of the wavefront that are unclear how to be treated. Having the eikonal equations involving the refractive indices and velocity components only would be of advantage. As a cross check with the previously obtained results the refractive indices can be derived from . For the isotropic case, using the second of we obtain $|\boldsymbol{\nabla}\psi|^2=n^2$, which by inserting into directly leads to the isotropic result $n|_{\circledcirc}=1/\mathcal{A}$. In of the anisotropic (nonbirefringent) sector we can introduce $$(\partial^1\psi)^2+(\partial^2\psi)^2=n^2\sin^2\vartheta\,,\quad\partial^3\psi=n\cos\vartheta\,,$$ leading to $n|_{\varobar}=1/\sqrt{\sin^2\vartheta+\mathcal{B}^2\cos^2\vartheta}$. The latter depends on the angle $\vartheta$ between the propagation direction and the preferred direction $\boldsymbol{\zeta}$. For the anisotropic (birefringent) sector we insert $$\partial^1\psi=n\frac{\cos\theta_1-\cos\theta_2}{2\sin\eta}\,,\quad \partial^3\psi=n\frac{\cos\theta_1+\cos\theta_2}{2\cos\eta}\,,\quad \partial^2\psi=\sqrt{n^2-(\partial^1\psi)^2-(\partial^3\psi)^3}\,,$$ both in and to obtain two refractive indices differing at second order in Lorentz violation: $$\begin{aligned} n|_{\varovee}^{(1)}&=\sqrt{\frac{1-\mathcal{G}^2}{1+\mathcal{G}\left\{(\mathcal{G}/2)\left[\cos(2\theta_1)+\cos(2\theta_2)\right]-2\cos\theta_1\cos\theta_2\right\}}}\,, \displaybreak[0]\\[2ex] n|_{\varovee}^{(2)}&=\sqrt{\frac{1+\mathcal{G}\cos(2\eta)}{1+\mathcal{G}[\cos(2\eta)-2\cos\theta_1\cos\theta_2]}}\,.\end{aligned}$$ Based on these refractive indices the integrands of the action functional in can be computed. The results are consistent with Eqs. (\[eq:finsler-structure-isotropic\]), (\[eq:finsler-structure-anisotropic\]): $$\begin{aligned} V(\mathbf{u})|_{\circledcirc}&=n|_{\circledcirc}|\mathbf{u}|=\frac{1}{\mathcal{A}}\sqrt{\mathbf{u}^2}=F(\mathbf{u})|_{\circledcirc}^+\,, \displaybreak[0]\\[2ex] V(\mathbf{u})|_{\varobar}&=n|_{\varobar}|\mathbf{u}|=\frac{\sqrt{(u^1)^2+(u^2)^2+(u^3)^2}}{\sqrt{\sin^2\vartheta+\mathcal{B}^2\cos^2\vartheta}}=\frac{(u^1)^2+(u^2)^2+(u^3)^2}{\sqrt{(u^1)^2+(u^2)^2+\mathcal{B}^2(u^3)^2}} \notag \\ &=F(\mathbf{u})|_{\varobar}^+\,, \displaybreak[0]\\[2ex] V(\mathbf{u})|_{\varovee}^{(1)}&=n|_{\varovee}^{(1)}|\mathbf{u}|=F(\mathbf{u})|_{\varovee}^{(1)+}\,,\quad V(\mathbf{u})|_{\varovee}^{(2)}=n|_{\varovee}^{(2)}|\mathbf{u}|=F(\mathbf{u})|_{\varovee}^{(2)+}\,,\end{aligned}$$ where for the latter two has to be employed. The refractive indices obtained from the wavefront equations correspond to the refractive indices computed directly from their definitions via the inverse phase velocity: $n\equiv v_{\mathrm{ph}}^{-1}=|\mathbf{k}|/\omega$. $$\begin{aligned} \label{eq:refraction-index-isotropic} n|_{\circledcirc}&=v_{\mathrm{ph}}|_{\circledcirc}^{-1}=\frac{|\mathbf{k}|}{\omega|_{\circledcirc}}=\frac{1}{\mathcal{A}}\,, \\[2ex] \label{eq:refraction-index-anisotropic} n|_{\varobar}&=v_{\mathrm{ph}}|_{\varobar}^{-1}=\frac{|\mathbf{k}|}{\omega|_{\varobar}}=\sqrt{\frac{\mathbf{k}^2}{k_{\bot}^2+\mathcal{B}^2k_{\scalebox{0.6}{$\|$}}^2}}=\frac{1}{\sqrt{\sin^2\vartheta+\mathcal{B}^2\cos^2\vartheta}}\,, \\[2ex] \label{eq:refraction-index-anisotropic-birefringent} n|_{\varovee}^{(1)}&=(v_{\mathrm{ph}}|_{\varovee}^{(1)})^{-1}=\frac{|\mathbf{k}|}{\omega_1|_{\varovee}}\,,\quad n|_{\varovee}^{(2)}=(v_{\mathrm{ph}}|_{\varovee}^{(2)})^{-1}=\frac{|\mathbf{k}|}{\omega_2|_{\varovee}}\,.\end{aligned}$$ These previously performed studies do not reveal any inconsistencies. The essential conclusion is that it should be warranted to describe the isotropic, anisotropic (nonbirefringent), and anisotropic (birefringent) sectors of modified Maxwell theory (in the geometric-optics approximation) with an adapted version of the eikonal equation, . Last but not least the parity-odd sector of shall be elaborated on. The wavefront equations for the parity-odd case were not derived in [@Xiao:2010yx], since in the latter reference all controlling coefficients mixing electric and magnetic fields were set to zero. Adapting the procedure used allows to derive them nevertheless. The authors of [@Xiao:2010yx] consider the values of the fields directly on the wavefront, e.g., for the electric field: $\mathbf{E}_0(\mathbf{x})=\mathbf{E}(t,\mathbf{x})|_{t=\psi(\mathbf{x})}$. In what follows, all fields evaluated on the wavefront will be denoted by an additional “0” as an index. The spatial derivative on the wavefront is then given by: $$\frac{\partial \mathbf{E}_0}{\partial x^j}=\frac{\partial \mathbf{E}}{\partial x^j}+\dot{\mathbf{E}}\frac{\partial\psi}{\partial x^j}\,.$$ Based on this procedure, from Maxwell’s equations four equations can be derived that involve field components on the wavefront and field derivatives only: $$\begin{aligned} \boldsymbol{\nabla}\times \mathbf{E}_0&=-\dot{\mathbf{B}}+\boldsymbol{\nabla}\psi \times \dot{\mathbf{E}}\,,\quad \boldsymbol{\nabla}\times \mathbf{H}_0=\dot{\mathbf{D}}+\boldsymbol{\nabla}\psi\times \dot{\mathbf{H}}\,, \\[2ex] \boldsymbol{\nabla}\cdot \mathbf{D}_0&=\boldsymbol{\nabla}\psi\cdot \dot{\mathbf{D}}\,,\quad \boldsymbol{\nabla}\cdot \mathbf{B}_0=\boldsymbol{\nabla}\psi\cdot \dot{\mathbf{B}}\,, \\[2ex] \mathbf{D}&=\mathbf{E}+\kappa_{\scriptscriptstyle{DB}}\mathbf{B}\,,\quad \mathbf{H}=\kappa_{\scriptscriptstyle{DB}}\mathbf{E}+\mathbf{B}\,.\end{aligned}$$ These must be combined to obtain an equation that involves the time derivatives of only a single field, e.g., the electric field and field values on the wavefront that may not necessarily include only a single field. This can be carried out via the following chain of steps: $$\begin{aligned} \boldsymbol{\nabla}\times \mathbf{E}_0&=-\dot{\mathbf{H}}+\kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi \times \dot{\mathbf{E}}\,, \displaybreak[0]\\[2ex] \boldsymbol{\nabla}\psi \times (\boldsymbol{\nabla}\times \mathbf{E}_0)&=-\boldsymbol{\nabla}\psi\times \dot{\mathbf{H}}+\boldsymbol{\nabla}\psi \times \kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi\times(\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}})\,, \displaybreak[0]\\[2ex] \boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\times \mathbf{E}_0)&=\dot{\mathbf{D}}-\boldsymbol{\nabla}\times \mathbf{H}_0+\boldsymbol{\nabla}\psi\times \kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}})\,, \displaybreak[0]\\[2ex] \boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\times \mathbf{E}_0)&=\dot{\mathbf{E}}+\kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{B}}-\boldsymbol{\nabla}\times \mathbf{H}_0+\boldsymbol{\nabla}\psi\times \kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}})\,, \displaybreak[0]\\[2ex] \boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\times \mathbf{E}_0)&=\dot{\mathbf{E}}+\kappa_{\scriptscriptstyle{DB}}(\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}}-\boldsymbol{\nabla}\times \mathbf{E}_0)-\boldsymbol{\nabla}\times \mathbf{H}_0 \notag \\ &\phantom{{}={}}+\boldsymbol{\nabla}\psi\times \kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}})\,.\end{aligned}$$ The resulting equation then reads $$\begin{aligned} \dot{\mathbf{E}}+\kappa_{\scriptscriptstyle{DB}}\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}}+\boldsymbol{\nabla}\psi \times \kappa_{\scriptscriptstyle{DB}}\dot{\mathbf{E}}+\boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\psi\times \dot{\mathbf{E}})&=\boldsymbol{\nabla}\psi\times (\boldsymbol{\nabla}\times \mathbf{E}_0)+\boldsymbol{\nabla}\times \mathbf{H}_0 \notag \\ &\phantom{{}={}}+\kappa_{\scriptscriptstyle{DB}}\boldsymbol{\nabla}\times \mathbf{E}_0\,.\end{aligned}$$ The condition for a vanishing determinant of the matrix on the left-hand side for the existence of nontrivial solutions leads to the wavefront equation for the parity-odd case. For consistency we pull the index of $\boldsymbol{\nabla}\psi$ down: $$\label{eq:eikonal-equation-parity-odd} \left(1-2\boldsymbol{\zeta}\cdot \boldsymbol{\nabla}\psi-|\boldsymbol{\nabla}\psi|^2\right)\left[1-(1+\boldsymbol{\zeta}^2)|\boldsymbol{\nabla}\psi|^2-2(\boldsymbol{\zeta}\cdot\boldsymbol{\nabla}\psi)+(\boldsymbol{\zeta}\cdot\boldsymbol{\nabla}\psi)^2\right]=0\,.$$ Inserting the second of in the first factor of results in $$1-2\boldsymbol{\zeta}\cdot \boldsymbol{\nabla}\psi-|\boldsymbol{\nabla}\psi|^2=1-2n\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}}-n^2 \overset{!}{=} 0\,.$$ The latter can be solved with respect to the refractive index $n$ to give $$\label{eq:refraction-index-parity-odd-1} n|_{\otimes}^{\boldsymbol{\zeta}(1)}=-\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}}+\sqrt{1+(\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}})^2}=-\mathcal{E}\cos\vartheta+\sqrt{1+\mathcal{E}^2\cos^2\vartheta}\,,$$ where only the positive-sign solution delivers a physically meaningful refractive index. Hence the result obtained from the eikonal equation is consistent with , which can be seen upon close inspection: $$V(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(1)}=n|_{\otimes}^{\boldsymbol{\zeta}(1)}|\mathbf{u}|=-\boldsymbol{\zeta}\cdot\mathbf{u}+\sqrt{\mathbf{u}^2+(\boldsymbol{\zeta}\cdot\mathbf{u})^2}=F(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(1)+}\,.$$ The same procedure applied to the second factor of leads to: $$1-(1+\mathcal{E}^2)n^2-2n(\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}})+n^2(\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}})^2=0\,.$$ Therefore the refractive index reads $$\label{eq:refraction-index-parity-odd-2} n|_{\otimes}^{\boldsymbol{\zeta}(2)}=\frac{-\boldsymbol{\zeta}\cdot \widehat{\mathbf{u}}+\sqrt{1+\mathcal{E}^2}}{1+\mathcal{E}^2-(\boldsymbol{\zeta}\cdot\widehat{\mathbf{u}})^2}=\frac{-\mathcal{E}\cos\vartheta+\sqrt{1+\mathcal{E}^2}}{1+\mathcal{E}^2\sin^2\vartheta}\,,$$ which is consistent with $$V(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(2)}=n|_{\otimes}^{\boldsymbol{\zeta}(2)}|\mathbf{u}|=\frac{-\boldsymbol{\zeta}\cdot\mathbf{u}+\sqrt{1+\mathcal{E}^2}|\mathbf{u}|}{1+\mathcal{E}^2-(\boldsymbol{\zeta}\cdot\mathbf{u})^2/\mathbf{u}^2}=F(\mathbf{u})|_{\otimes}^{\boldsymbol{\zeta}(2)+}\,.$$ The refractive indices obtained from the wavefront equations for the isotropic and anisotropic cases, Eqs. (\[eq:eikonal-equations-isotropic\]) – (\[eq:eikonal-equations-anisotropic-birefringent-1\]), respectively, are consistent with the usual definition of the refractive index via the inverse phase velocity (cf. Eqs (\[eq:refraction-index-isotropic\]) – (\[eq:refraction-index-anisotropic-birefringent\]). However this does not seem to be the case for the parity-odd sector. Inspecting Eqs. (\[eq:dispersion-relation-parity-odd-1\]), (\[eq:dispersion-relation-parity-odd-2\]) and the latter results for the refractive indices of Eqs. (\[eq:refraction-index-parity-odd-1\]), (\[eq:refraction-index-parity-odd-2\]) reveals the inconsistency: $$\begin{aligned} \frac{|\mathbf{k}|}{\omega_1|_{\otimes}}&=\frac{1}{\mathcal{E}\cos\vartheta+\sqrt{1+\mathcal{E}^2\cos^2\vartheta}}\neq n|_{\otimes}^{\boldsymbol{\zeta}(1)}\,, \\[2ex] \frac{|\mathbf{k}|}{\omega_2|_{\otimes}}&=\frac{1}{\mathcal{E}\cos\vartheta+\sqrt{1+\mathcal{E}^2}}\neq n|_{\otimes}^{\boldsymbol{\zeta}(2)}\,.\end{aligned}$$ The definition of the refractive index via the inverse of the phase velocity rests on the existence of a nonzero permeability and permittivity. However for the parity-odd case they both vanish and the electric fields even mix with the magnetic fields, which is why the ordinary definition of the refractive index does not seem to be reasonable. A further origin of the issue may be that Okubo’s method does not produce Finsler structures in a unique manner. We conclude that it may be problematic to treat the parity-odd case of modified Maxwell theory with the eikonal equation. Finding a solution to this clash is an interesting open problem. Gravitational backgrounds {#sec:gravitational-backgrounds} ========================= The physics for a classical point-particle equivalent to a massive fermion rests on its Lagrangian. The procedure of deriving those within the framework of the SME works for massive particles only where in the limit of a vanishing particle mass the Lagrangian goes to zero. So far we have demonstrated that the important quantity to describe the physics of electromagnetic waves in the geometric-optics approximation is the refractive index. The reason is that the motion of photons is much more restricted than the motion of a massive particle. After all, for a particle with mass moving in a potential the initial position, direction, and velocity can be chosen freely. On the contrary, for a photon the initial position and direction only are not fixed, whereas its initial speed is determined by the refractive index at its starting point. In the previous sections it was shown how to establish connections between various cases of the minimal SME photon sector and certain Finsler geometries. The Finsler geometries found were discovered to be closely related to the various refractive indices where only for the parity-odd case of the [*CPT*]{}-even sector such a connection is not manifest. The refractive indices found are independent of the spacetime position such as the controlling coefficients, which corresponds to the analogue of a homogeneous medium in optics. However the refractive index can depend on the three spatial velocity components. In other words, in such cases the refractive index depends on angles enclosed between the propagation direction and preferred directions. This situation is reminiscent of anisotropic media in optics. Hence Finsler structures related to the [*CPT*]{}-even photon sector are three-dimensional in contrast to the Finsler structures obtained from Wick-rotating classical Lagrangians of massive particles. Besides, note that in the photon case no Wick rotation is necessary, since the intrinsic metric involved is already of Euclidean signature. These results shall serve as a base to study light rays in the geometric-optics approximation in the presence of Lorentz violation. As we saw, for most cases these can be described by the eikonal equation, cf. : $$\frac{\mathrm{d}}{\mathrm{d}s}\left[n\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\right]=\boldsymbol{\nabla}_{\mathbf{x}}n\,,$$ where $n$ is the refractive index of the medium considered. On the right-hand side the gradient is understood to be computed with respect to the position vector $\mathbf{x}$. The photon trajectory is given by $\mathbf{x}=\mathbf{x}(s)$ and it is parameterized by the arc length $s$. For an isotropic and homogeneous medium the refractive index is a mere constant. In this case one immediately sees that the resulting ray equation is $$\label{eq:eikonal-equation-homogeneous-medium} \frac{\mathrm{d}^2\mathbf{x}}{\mathrm{d}s^2}=\mathbf{0}\,,$$ whose solution is a straight line as expected. For homogeneous, but anisotropic media the refractive index depends on at least one angle, $n=n(\vartheta)$, where further angles are suppressed for brevity. For a straight ray trajectory the angle $\vartheta$ is fixed by the initial direction and it does not change during propagation, i.e., it is not a function of $s$. Furthermore due to homogeneity the refractive index does not change along the trajectory as well, which is why $\boldsymbol{\nabla}_{\mathbf{x}}n(\vartheta)=0$ for points on the trajectory. Therefore in this case we again end up with . For inhomogeneous media with $n=n(\mathbf{x})$ the eikonal equation cannot have straight-line solutions, though. In what follows the formalism and knowledge attained shall be applied to propagating light rays in curved spacetimes with metric $g_{\mu\nu}=g_{\mu\nu}(x)$. The trajectory of a ray in a spacetime is described by a four-vector $x^{\mu}=x^{\mu}(s)$ and it propagates with the four-velocity $u^{\mu}\equiv \mathrm{d}x^{\mu}/\mathrm{d}s$. Propagation occurs along geodesics combined with the nullcone condition $g_{\mu\nu}u^{\mu}u^{\nu}=0$ that has to hold locally at each spacetime point. For practical reasons, which will become clear in the course of the current section, all forthcoming investigations will be performed in a spacetime characterized by a line interval of the form $$\label{eq:line-interval} \mathrm{d}\tau^2=\frac{1}{A(r,\theta,\phi)}\mathrm{d}t^2-A(r,\theta,\phi)(\mathrm{d}r^2+r^2\mathrm{d}\theta^2+r^2\sin^2\theta\mathrm{d}\phi^2)\,.$$ Here $t$ is the time, $(r,\theta,\phi)$ are spherical coordinates, and $A$ is a time-independent function. Such metrics were proposed in [@Wu:1988] and they are denoted as “generally isotropic” where metrics with $A=A(r)$ are called “spherically symmetric.” The parentheses in the spatial part of give the volume element of a three-dimensional ball and it is multiplied by $A(r,\theta,\phi)$. The choice $A(r,\theta,\phi)=1$ in describes Minkowski spacetime in three-dimensional spherical coordinates. In this case the spatial coordinate surfaces with constant $r$ are two-spheres. For arbitrary $A(r,\theta,\phi)$ these surfaces are still two-spheres topologically, but their local geometry depends on $r$, $\theta$, and $\phi$. Note that the metric describing a weak gravitational field can be brought into the generally isotropic form: $$\begin{aligned} \label{eq:metric-weak-gravitational-field} (g_{\mu\nu})&=\mathrm{diag}\Big((1+2\Phi),-(1-2\Phi),-(1-2\Phi),-(1-2\Phi)\Big) \notag \\ &=\mathrm{diag}\Big(\frac{1}{1-2\Phi},-(1-2\Phi),-(1-2\Phi),-(1-2\Phi)\Big)+\mathcal{O}(\Phi^2)\,.\end{aligned}$$ Here $\Phi=\Phi(r)=-GM/r\ll 1$ is the Newtonian potential. In the latter paper [@Wu:1988] it was shown that there is a link between the eikonal equation of the geometric-optics approximation and the null geodesic equations of a spacetime based on a line interval of . A suitable combination of the geodesic equations leads to $$\frac{\mathrm{d}}{\mathrm{d}s}\left[A(r,\theta,\phi)\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\right]=\boldsymbol{\nabla}A(r,\theta,\phi)\,,$$ i.e., $A(r,\theta,\phi)$ of can be understood as an inhomogeneous and anisotropic refractive index. Therefore as long as weak gravitational fields are considered, light behaves according to the geometric-optics approximation. The approximation is expected to break down as soon as strong gravitational forces appear such as in the direct vicinity of a black hole. In this case the original geodesic equations have to be studied instead of the eikonal approach. Note that the converse is true as well. If the eikonal equation is known to be valid (also in flat spacetime) this corresponds to a propagating ray in a generally isotropic spacetime of . Isotropic case {#sec:gravitational-backgrounds-isotropic-case} -------------- The eikonal approach has a great potential to be applied to the propagation of light rays in a weak gravitational field permeated by a Lorentz-violating background field. It is reasonable to start with the simplest case, which is the isotropic one investigated in . With the constant refractive index $n=1/\mathcal{A}$ (in Minkowski spacetime) given by or the eikonal equation and the corresponding spacetime, , read as follows: $$\begin{aligned} \label{eq:line-interval-isotropic-case} \frac{\mathrm{d}}{\mathrm{d}s}\left[\frac{1}{\mathcal{A}}\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\right]&=\boldsymbol{\nabla}\left(\frac{1}{\mathcal{A}}\right)\,, \\[2ex] \mathrm{d}\tau^2&=\mathcal{A}\mathrm{d}t^2-\frac{1}{\mathcal{A}}(\mathrm{d}r^2+r^2\mathrm{d}\theta^2+r^2\sin^2\theta\mathrm{d}\phi^2)\,.\end{aligned}$$ The coordinate surfaces of the associated spacetime are spheres whose radii are scaled by $1/\sqrt{\mathcal{A}}$. This intermediate result can now be used to introduce a gravitational background. Via the principle of minimal coupling the flat Minkowski metric is replaced by a curved spacetime metric, $\eta_{\mu\nu}\mapsto g_{\mu\nu}(x)$, and the constant refractive index $n$ is promoted to a spacetime-position dependent function: $n\mapsto n(r,\theta,\phi)$. The curved spacetime metric is taken to be for a weak gravitational field. Since the latter is spherically symmetric, it is reasonable to assume spherical symmetry for the position-dependent refractive index, i.e., $n(r)=1/\mathcal{A}(r)$. The corresponding eikonal equation and the line interval then read as follows: $$\begin{aligned} \label{eq:refractive-index-isotropic-gravitational} \frac{\mathrm{d}}{\mathrm{d}s}\left[n(r)\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}s}\right]&=\boldsymbol{\nabla}n(r)\,,\quad n(r)\equiv \frac{1-2\Phi(r)}{\mathcal{A}(r)}\,, \\[2ex] \label{eq:spherical-metric-isotropic-modmax} \mathrm{d}\tau^2&=\frac{1}{n(r)}\mathrm{d}t^2-n(r)(\mathrm{d}r^2+r^2\mathrm{d}\theta^2+r^2\sin^2\theta\mathrm{d}\phi^2)\,.\end{aligned}$$ ![Deflection of light near a massive body, e.g., the planet Jupiter. (The picture of Jupiter was taken by the Cassini spacecraft, cf. <http://solarsystem.nasa.gov/planets/profile.cfm?Object=Jupiter>.)[]{data-label="fig:light-deflection"}](deflection-angle.pdf) Hence the minimal-coupling principle amounts to a refractive index that is the product of a spatial component of the weak gravitational field metric and the spacetime-position dependent refractive index $1/\mathcal{A}(r)$ associated to the isotropic Lorentz-violating framework considered. The approach introduced has a paramount advantage. The physics of a Lorentz-violating photon in a (weak) gravity field can be studied without field theory and the geodesic equations in a curved spacetime. Instead, a classical method is used replacing photons by light rays and working in the geometric-optics approximation with the eikonal equation. In this context Lorentz symmetry violation is treated as explicit, which is known to clash with the existence of gravitational backgrounds [@Kostelecky:2003fs]. The latter sections \[sec:modified-energy-momentum-conservation\] and \[sec:properties-isotropic-finsler\] will be dedicated to this issue where for now we will delve into phenomenology. One possible application of the used approach lies in the (modified) deflection of light in the vicinity of a massive body (cf. ), which is an important test of gravitational theories. From a technical point of view the eikonal equation is nonlinear, which makes it challenging to solve analytically in general. However for the isotropic case, i.e., a refractive index only depending on the radial coordinate $r$ the formula of Bouguer follows from the eikonal equation (see, e.g., Sec. 3.2.1 of [@Born:1999]): $$\label{eq:bouguer-formula-nonintegrated} n(r)r\sin\alpha=C\,.$$ Here $C$ is a constant and $\alpha$ the angle between the tangent vector of the trajectory and the radial vector pointing from the coordinate origin to a particular point on the trajectory. The latter relationship is the equivalent of energy and angular momentum conservation for a massive particle in classical mechanics. Since both the distance $r$ of a particular point from the origin and the angle $\alpha$ associated to this point does not depend on the parameterization of the trajectory, we choose to parameterize it by spherical coordinates. Thereby the problem is restricted to the $x$-$z$-plane with $\theta=\pi/2$. The trajectory then reads $\mathbf{x}=r\widehat{\mathbf{e}}_r$ where $r=r(\phi)$ and $\widehat{\mathbf{e}}_r=\widehat{\mathbf{e}}_r(\phi)$ is the unit vector pointing in radial direction. The angle $\alpha$ is given as follows: $$\sin\alpha=\frac{r(\phi)}{\sqrt{r^2(\phi)+\dot{r}^2(\phi)}}\,,\quad \dot{r}\equiv \frac{\mathrm{d}r}{\mathrm{d}\phi}\,.$$ Now the formula of Bouguer delivers a differential equation for $\phi(r)$. Its solution is obtained by solving the latter with respect to $\mathrm{d}\phi/\mathrm{d}r$ and by performing a subsequent integration: $$\label{eq:integration-deflection-angle} \phi(r)=C\int_d^r \frac{\mathrm{d}r}{r\sqrt{n(r)^2r^2-C^2}}\,,$$ where $d$ is the distance of minimal proximity and the condition $\phi(d)=0$ has been set. By doing so, solving the eikonal equation has been reduced to computing a one-dimensional integral. Now consider a classical light ray approaching a massive body with impact parameter $d_{\infty}$, which is the distance between the particle propagation direction in the asymptotically flat region and the parallel going through the center of mass of the body (at the coordinate origin). The photon will travel such that its distance to the body steadily decreases until reaching a minimum where it increases again afterwards. At the minimum distance $d$ we have that $\dot{r}=0$ and therefore $\alpha=\pi/2$. The minimum distance corresponds to the impact parameter to a very good approximation: $d\approx d_{\infty}$. This is why immediately tells us that $$C=n(d)d\approx n(d_{\infty})d_{\infty}\,.$$ Without the massive body the change $\Delta\phi$ in the angle would be equal to $\pi$ for a photon coming from an asymptotically flat region, passing near the coordinate origin, and propagating back to infinity. Due to the body there is a deflection, which changes $\Delta\phi$ to an angle that is slightly larger than $\pi$. Performing the integration in from $r=d$ to infinity gives half of this contribution, since it only takes into account the second half of the trajectory. Therefore the deflection angle $\varphi$ is given by $$\label{eq:deflection-angle} \varphi=\Delta\phi-\pi=2C\int_d^{\infty} \frac{\mathrm{d}r}{r\sqrt{n(r)^2r^2-C^2}}-\pi\,.$$ It can be checked that gives $\varphi=0$ for $n(r)=1$ as expected. For a constant refractive index $n$ it holds that $C=nd$. By inspecting it follows immediately that a constant $n$ does not lead to any deflection. This is in contrast to [@Betschart:2008yi] where for certain Lorentz-violating frameworks with [*constant*]{} Lorentz-violating coefficient it was shown that there is a change in the deflection angle caused by Lorentz violation, indeed. However note that in the latter reference a Schwarzschild black hole was considered whose line interval had not been cast into generally isotropic form, cf. . A discussion of this difference leading to more insight into Bouguer’s formula is relegated to , since it is quite technical and probably not of relevance for all readers. Phenomenology for the isotropic framework ----------------------------------------- With the technique further developed, we are ready to carry out phenomenological calculations. The goal is to obtain predictions for the change of the light deflection angle caused by particular Lorentz-violating frameworks. These predictions will be compared to experiment to obtain sensitivities on controlling coefficients in the minimal SME photon sector. As the most important example light deflection at the Sun will be discussed first. However light can be deflected at any other massive bodies such as planets. First of all we intend to recapitulate the standard result. For vanishing Lorentz violation the deflection angle of can be computed analytically. Thereby the integral 2.266 of [@Gradshteyn:2007] is helpful: $$\label{eq:gradshteyn-integral} \int \frac{\mathrm{d}x}{x\sqrt{\alpha+\beta x+\gamma x^2}}=\frac{1}{\sqrt{-\alpha}}\arcsin\left(\frac{2\alpha+\beta x}{x\sqrt{\beta^2-4\alpha\gamma}}\right)\,,\quad \alpha<0\,,\quad \beta^2-4\alpha\gamma>0\,.$$ For the Lorentz-invariant case we have $$\alpha=-\frac{d}{R_S}\left(\frac{d}{R_S}+2\right)\,,\quad \beta=2\,,\quad \gamma=1\,,$$ with the Schwarzschild radius $R_S=2GM/c^2$ of the massive body. Here $G$ is the gravitational constant, $M$ the mass of the body, and $c$ the speed of light. The conditions for $\alpha$, $\beta$, and $\gamma$ stated in are fulfilled and the full analytical result for the deflection angle is given as follows: $$\label{eq:standard-light-deflection} \varphi=\frac{1+2\xi}{\sqrt{1+4\xi}}\left[\pi+2\arcsin\left(\frac{2\xi}{1+2\xi}\right)\right]-\pi=4\xi+\mathcal{O}(\xi^2)\,,\quad \xi=\frac{R_S}{2d}\,,$$ where the latter is the first-order expansion in the dimensionless parameter $\xi\ll 1$. Now considering a light ray directly passing the surface of the Sun (scraping incidence), $d$ is given by the radius $r_{\scriptscriptstyle{\bigodot}}$ of the Sun. Using the values of and multiplying the previous equation with $180\cdot 60^2/\pi$ leads to the well-known result $\varphi\approx 1.75''$, which lies within few standard deviations from the mean value observed during the total eclipse in 1919 [@Dyson:1919; @Eddington:1923]. [ccc]{} Quantity & Unit & Value\ $G$ & $\mathrm{m^3/(kg\cdot s^2)}$ & $6.67384\cdot 10^{-11}$\ $M_{\astrosun}$ & kg & $1.98910\cdot 10^{30}$\ $M_{\jupiter}$ & kg & $1.89813\cdot 10^{27}$\ $M_{\saturn}$ & kg & $5.68319\cdot 10^{26}$\ $r_{\scriptscriptstyle{\astrosun}}$ & m & $6.95508\cdot 10^8$\ $r_{\scriptscriptstyle{\jupiter}}$ & m & $6.99110\cdot 10^7$\ $r_{\scriptscriptstyle{\saturn}}$ & m & $5.82320\cdot 10^7$\ ![Spacetime-position dependent refractive index (and controlling coefficient) as a function of the dimensionless parameter $r/d$ where $d$ corresponds to the Sun radius $r_{\scriptscriptstyle{\bigodot}}$ in this example. (The picture of the Sun was taken by SOHO – EIT Consortium, ESA, NASA, cf. <http://science.nasa.gov/science-news/science-at-nasa/2003/22apr_currentsheet>.)[]{data-label="fig:position-dependent-controlling-coefficient"}](sample-functions-controlling-coefficients.pdf) Now the refractive index is modified due to Lorentz violation according to . Therefore the isotropic Lorentz-violating coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ is promoted to a spacetime-position dependent function (cf. ). It is assumed to only depend on the radial coordinate $r$ to keep the framework isotropic:$$\widetilde{\kappa}_{\mathrm{tr}}\mapsto \widetilde{\kappa}_{\mathrm{tr}}(r)=\widetilde{\kappa}_{\mathrm{tr}}\left[1-f(r)\right]\,,$$ with a function $f$ having special properties. The latter shall be constructed such that $1-f\geq 0$ for $r/d\geq 1$. This means that the sign of $\widetilde{\kappa}_{\mathrm{tr}}(r)$ is fixed by the sign of the constant prefactor $\widetilde{\kappa}_{\mathrm{tr}}$. Furthermore $\lim_{r\mapsto\infty} \widetilde{\kappa}_{\mathrm{tr}}(r)=\widetilde{\kappa}_{\mathrm{tr}}$, whereby in the asymptotically flat region the position-dependent controlling coefficient is identified with the corresponding SME photon coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ in Minkowski spacetime. The refractive index then reads as $$\label{eq:refractive-index-modelling} n(r)=\sqrt{\frac{1+\widetilde{\kappa}_{\mathrm{tr}}(r)}{1-\widetilde{\kappa}_{\mathrm{tr}}(r)}}\left(1+\frac{R_S}{r}\right)\,.$$ From the coordinate velocity of light in this framework is given by $$c=\frac{|\mathrm{d}\mathbf{r}|}{\mathrm{d}t}=\frac{1}{n(r)}\,,$$ i.e., for $\widetilde{\kappa}_{\mathrm{tr}}>0$ it is reduced in comparison to the Lorentz-invariant case. The position dependence shall reflect the properties of the gravitational background. The curvature radius $R_S$ is the physical scale of the background, i.e., it is reasonable to associate it with $\widetilde{\kappa}_{\mathrm{tr}}(r)$ as well. Note that we are only interested in the behavior of the function outside of the massive body, which means $r\geq d$. Generic functions with these properties are \[eq:sample-functions\] $$\begin{aligned} \label{eq:sample-functions-f} f(r)&\equiv \left[1+a\left(\frac{r-d}{R_S}\right)^2\right]^{-1}\,, \\[2ex] \label{eq:sample-functions-g} g(r)&\equiv \frac{2\arctan\left\{a\left[1-(r-d)^2/R_S^2\right]\right\}+\pi}{2\arctan(a)+\pi}\,, $$ where $a\leq 1$ is a free, dimensionless parameter. Therefore for these particular sample functions it holds that $f(d)=1$ and $\lim_{r\mapsto\infty} f(r)=0$. Whatever the underlying theory for a possible violation of Lorentz invariance looks like, it is reasonable to assume that the amount of Lorentz violation is influenced by a gravitational background field. Referring to a small-scale structure of spacetime where simple models were shown to produce Lorentz-violating particle dispersion relations [@Klinkhamer:2003ec; @Bernadotte:2006ya] the argument could be along the following lines. A gravitational field has an energy density associated to it, cf. [@Lynden-Bell:1985] for the case of spheres and black holes. Since a spacetime foam is caused by energy fluctuations, an additional contribution of energy density associated to a gravitational field may have some influence on it. This would render the effective controlling coefficients for Lorentz violation spacetime-position dependent. Hence for the isotropic framework considered the refractive index directly at the surface of the Sun may have a dip for $\widetilde{\kappa}_{\mathrm{tr}}>0$ or a peak for $\widetilde{\kappa}_{\mathrm{tr}}<0$ in its position dependence (cf. ). As long as the underlying description is not available, it is challenging to deliver a more rigorous argumentation. Hence a $\widetilde{\kappa}_{\mathrm{tr}}(r)$ including with the parameter $a$ controlling the width of the dip/peak must be interpreted as a phenomenological description of such effects. Now the modified deflection angle can be calculated in two different ways. The first is to compute the integral according to Bouguer’s formula of . The second is to solve the eikonal equation directly. In the eikonal equation is brought into a form that is suitable for solving it. For a refractive index that has a radial dependence only, results in $$\label{eq:eikonal-equation-isotropic-numerical-basis} 0=(r^2+\dot{r}^2)r\frac{\partial n}{\partial r}+n(r^2+2\dot{r}^2-r\ddot{r})\,.$$ Bouguer’s formula is a first integral of the eikonal equation that follows from angular momentum conservation. Therefore using it allows us to avoid the computation of one integral. Nevertheless as a cross check it is reasonable to carry out the computation with the two techniques. Both the integral of and the eikonal equation are challenging to be solved analytically for a refractive index that is modified by Lorentz violation. Therefore we attempt to treat both cases numerically with `Mathematica`. To gain some physical understanding, the eikonal equation is solved numerically for hypothetical values of $R_S$ and $\widetilde{\kappa}_{\mathrm{tr}}$ first. At the distance of minimal proximity $d$ the sample functions of vanish by construction. Therefore $n(d)=1+R_S/d$, which is why $C=n(d)d$ and the impact parameter is given by $$d_{\infty}=\frac{C}{n(r=\infty)}=\sqrt{\frac{1-\widetilde{\kappa}_{\mathrm{tr}}}{1+\widetilde{\kappa}_{\mathrm{tr}}}}\,(d+R_S)\,.$$ For realistic situations, i.e., light bending at stars the Schwarzschild radius is much smaller than the distance of minimal proximity. Note that for scraping incidence, $d$ corresponds to the radius of the star. Since bounds on the isotropic coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ in flat, asymptotic spacetime are strict, it holds that $d\approx d_{\infty}$ to a good approximation. For the hypothetical values that we choose for illustration purposes this is not necessarily warranted. Taking $d/R_S=5$ and $\widetilde{\kappa}_{\mathrm{tr}}=1/20$ we obtain the results depicted in . The curves show the solutions of the eikonal equation where the refractive index has been modeled according to using the sample function $f(r)$ of for different choices of the parameter $a$. Recall that the latter characterizes the width of the dip/peak in the refractive index directly at the surface of the massive body, which is caused by Lorentz violation. Since for comparison all curves should meet at a single point, the impact parameters $d_{\infty}$ have to be adapted properly, which is why they differ from each other. The observation is that for increasing $a$ and $\widetilde{\kappa}_{\mathrm{tr}}>0$ the deflection angle is reduced. As long as the light ray is far away from the massive body it experiences a refractive index that increases when the distance to the body decreases. This is the standard behavior of the refractive index whose origin lies in nonvanishing Riemann curvature components. Upon approaching the massive body the light ray suddenly experiences the dip where the refractive index becomes smaller for decreasing distance. The ray then behaves contrary to the standard case and tends to be bent away from the body, which can be clearly seen in . Note that for $\widetilde{\kappa}_{\mathrm{tr}}<0$ the dip in the refractive index turns into a peak. Hence the behavior is opposite and the ray is bent towards the body even stronger, cf. . From a technical point of view to solve the eikonal equation, proper initial conditions have to be considered. Since the ray is assumed to arrive from an asymptotically flat region, the initial angle is $\phi_0=\pi$. In practice an angle lying close to $\pi$ must be chosen where the direction of the ray initially is assumed to point along the positive horizontal axis. We express the solution of the eikonal equation as $r(\phi)=d\xi(\phi)$ with the dimensionless function $\xi(\phi)$. The initial conditions are then fixed to be $\xi(\phi_0)=\Delta$ and $\xi'(\phi_0)=-\cot(\phi_0)\Delta$ where $\phi_0=\pi-\arcsin(d_{\infty}/\Delta)$. Here $\Delta$ is a length scale with the property $\Delta \gg d$, which is tuned to increase the precision of the numerical result. Theoretically $\Delta$ should approach infinity, which is not a possible value to choose in practice, though. Setting the final angle in the numerical integration to $\phi_1\leq 0$ leads to numerical instabilities, which is why $\phi_1$ is taken to be slightly larger than zero. This is supposed to be sufficient for small bending angles that appear in realistic scenarios. It is reasonable to set both the working precision to a large number and the maximum number of steps to infinity. There are at least two space-based missions available that could test gravity based on light deflection. Two of the most promising ones are GAIA and LATOR. In what follows we will discuss the perspective of these missions in obtaining constraints on Lorentz violation in the (isotropic) photon sector by performing measurements of light deflection at massive bodies. Thereby the theoretical tools developed so far will be of great use. ### Sensitivity of GAIA GAIA[^2] [@Perryman:2001sp] is a space probe that was launched in December 2013 by ESA. The mission goal is to perform measurements of positions and radial velocities of about 1% of the galactic stellar population, which shall generate a three-dimensional map of our galaxy. This is supposed to give information on the galactic history, dark matter as well as extra-solar planetary systems. GAIA can measure angles with a sensitivity of around , which is why it can test deflection of light at massive bodies to a high precision. However the mission parameters do not allow light to be measured grazing the surface of the Sun. Such measurements will be possible for Jupiter and Saturn only (see Table III in [@Perryman:2001sp]). Now we intend to perform phenomenology of light bending in an isotropic Lorentz-violating framework based on the possibilities of GAIA. Thereby sample functions are taken according to with different values for the parameter $a=1/10^i$ and the range $i=0\dots 15$. Choosing a particular controlling coefficient $\widetilde{\kappa}_{\mathrm{tr}}$, the deflection angle of light in the vicinity of Jupiter is computed with two methods. The first uses the formula of Bouguer, . The second solves the eikonal equation (\[eq:eikonal-equation-isotropic-numerical-basis\]) numerically in analogy to what was described above. The bending angle is then computed via the scalar product of the initial and final normalized tangent vectors. This gives an excellent cross check for the results, since the two methods are independent from each other. [cc|c|c]{} & & $f(r)$ & $g(r)$\ $-\log_{10}(a)$ & $-\log_{10}(\widetilde{\kappa}_{\mathrm{tr}})$ & $\varphi_{\tiny{\jupiter}}^{*}-\varphi_{\tiny{\jupiter}}$ \[10arcs\] & $\varphi_{\tiny{\jupiter}}^{*}-\varphi_{\tiny{\jupiter}}$ \[10arcs\]\ 0 & 14 & 1.61 & 1.40\ $1\dots 4$ & 13 & 9.02; 5.07; 2.85; 1.60 & 8.80; 4.97; 2.80; 1.57\ $5\dots 8$ & 12 & 9.02; 5.07; 2.85; 1.60 & 8.84; 4.97; 2.80; 1.57\ $9\dots 12$ & 11 & 9.02; 5.07; 2.84; 1.59 & 8.84; 4.97; 2.79; 1.56\ $13\dots 15$ & 10 & 8.76; 4.67; 2.32 & 8.58; 4.57; 2.26\ The bending angle obtained is then compared to the standard result. This procedure is repeated for a decreasing isotropic coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ until the difference between the modified and the standard result approximately matches the precision that GAIA can measure angles with. This sets the sensitivity of the experiment with respect to $\widetilde{\kappa}_{\mathrm{tr}}$ in a curved background. However it is challenging to compute the integral or to solve the eikonal equation with a high precision. We use the difference of the results obtained from the two methods as a measure for how meaningful they are. For a conservative estimate of the sensitivity one should keep results only if this theoretical uncertainty is much smaller than the difference between the modified and the standard bending angle. First of all for $\widetilde{\kappa}_{\mathrm{tr}}>0$ the difference between the standard bending angle $\varphi_{\tiny{\jupiter}}^{*}$ and the modified bending angle is positive, which shows that the bending angle is reduced by a positive Lorentz-violating coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ (see the third column of ). For $\widetilde{\kappa}_{\mathrm{tr}}<0$ the behavior is vice versa and the absolute numbers mainly deviate in the third digit, which is why they are omitted in the table. We stated all differences $\varphi_{\tiny{\jupiter}}^{*}-\varphi_{\tiny{\jupiter}}$ that are larger than and lie in the vicinity of the experimental precision of GAIA, i.e., . Such modifications can be expected to be detectable by this mission. From the results it becomes clear that the sensitivity of the isotropic coefficient reduces when the width of the dip, which is controlled by the parameter $a$, decreases. If the width lies in the order of magnitude of Jupiter’s radius the sensitivity for $|\widetilde{\kappa}_{\mathrm{tr}}|$ is $10^{-14}$. In case the width lies 15 orders of magnitude below that the sensitivity of $|\widetilde{\kappa}_{\mathrm{tr}}|$ is still $10^{-10}$. Hence the sensitivity does not decrease as quickly as the parameter $a$. The numbers are meaningful, since the difference of the results obtained with Bouguer’s formula and by solving the eikonal equation directly is around at the maximum. The latter is interpreted as the theoretical uncertainty and it is much smaller than $|\varphi_{\tiny{\jupiter}}^{*}-\varphi_{\tiny{\jupiter}}|$. Obtaining the modified deflection angles for Saturn works completely analogously. The sensitivity on $\widetilde{\kappa}_{\mathrm{tr}}$ lies in the same order of magnitude. The only difference is that even smaller $a$ could be probed based on a modeling according to . The reason is that $$\frac{d_{\tiny\saturn}}{R_{S,\tiny\saturn}}\approx 2.78\frac{d_{\tiny\jupiter}}{R_{S,\tiny\jupiter}}\,,$$ whereby the additional dimensionless factor increases the contribution of $a$. ### Sensitivity of LATOR LATOR (Laser Astrometric Test of Relativity) [@Turyshev:2004ga; @Turyshev:2009zz] is a mission that is being planned by a collaboration of NASA and ESA. It is a Michelson-Morley-type experiment that shall perform curvature measurements in our solar system to determine the Eddington post-Newtonian parameter $\gamma$ with a precision of 1 part in $10^8$. It is considered to be a test mission for General Relativity and it is supposed to detect the frame-dragging effect and to determine the solar quadrupole moment. The primary objective will be to measure the gravitational deflection of light by the Sun to an accuracy of . Such an astounding precision shall be made possible by an improved laser ranging and a long-baseline optical interferometry system. [cc|c|c]{} & & $f(r)$ & $g(r)$\ $-\log_{10}(a)$ & $-\log_{10}(\widetilde{\kappa}_{\mathrm{tr}})$ & $\varphi_{\tiny{\astrosun}}^{*}-\varphi_{\tiny{\astrosun}}$ \[$10^{-2}$arcs\] & $\varphi_{\tiny{\astrosun}}^{*}-\varphi_{\tiny{\astrosun}}$ \[$10^{-2}$arcs\]\ 0 & 16 & 1.57 & —\ $(0)1\dots 4$ & 15 & 8.84; 4.97; 2.80; 1.57 & 13.6; 8.58; 4.84; 2.72; 1.53\ $5\dots 8$ & 14 & 8.84; 4.97; 2.79; 1.56 & 8.61; 4.84; 2.72; 1.52\ $9\dots 11$ & 13 & 8.57; 4.56; 2.26 & 8.35; 4.43; 2.18\ $12\dots 13$ & 12 & 9.99; 3.91 & 9.49; 3.64\ $14\dots 15$ & 11 & 13.9; 4.64 & 12.7; 4.21\ We carry out phenomenology as we did before by choosing different parameters $a$ for the sample functions of . The calculations are completely analogous to before where the only difference is that they are carried out for the Sun using the appropriate parameters of . The essential numerical results are stated in . The bending angle behaves similarly to before, i.e., it is reduced for $\widetilde{\kappa}_{\mathrm{tr}}>0$ and it increases for $\widetilde{\kappa}_{\mathrm{tr}}<0$. The differences $\varphi_{\tiny{\astrosun}}^{*}-\varphi_{\tiny{\astrosun}}$ are listed that lie in the vicinity of the experimental precision expected for LATOR, i.e., . If the width of the dip/peak in the refractive index of the Lorentz-violating vacuum lies in the order of magnitude of Sun’s radius the sensitivity for the isotropic coefficient $|\widetilde{\kappa}_{\mathrm{tr}}|$ is $10^{-16}$. The lowest sensitivity in case of a very narrow dip/peak is $10^{-11}$. Comparing the results determined from Bouguer’s formula to the results from the numerical solution of the eikonal equation reveals differences of ca. . Therefore the theoretical uncertainty is still much smaller than $|\varphi_{\tiny{\astrosun}}^{*}-\varphi_{\tiny{\astrosun}}|$. Note that for the model function $g(r)$ the modification of the deflection angle for $|\widetilde{\kappa}_{\mathrm{tr}}|=10^{-16}$ is smaller than $1.50\times 10^{-2}$arcs. Therefore assuming this model function, the sensitivity of LATOR will not be sufficient to detect a $|\widetilde{\kappa}_{\mathrm{tr}}|$ lying in the order of magnitude of $10^{-16}$. ### Discussion According to the current (2015) version of the data tables [@Kostelecky:2008ts] the strictest lower bounds on $\widetilde{\kappa}_{\mathrm{tr}}$ lie in the order of magnitude of $-10^{-16}$ where the best upper bounds are around $10^{-20}$. The isotropic coefficient of modified Maxwell theory is challenging to be constrained in laboratory experiments, which is why these bounds are related to ultra-high energy cosmic rays. With the precision of LATOR there would be a space-based experiment performed under controlled conditions that could have a sensitivity comparable to the best current constraints on a negative $\widetilde{\kappa}_{\mathrm{tr}}$. This is astonishing taking into account that the precision of a man-made experiment may match the sensitivity reached by the most energetic particles propagating through interstellar space for distances of many lightyears. It illustrates the versatility of the technique presented to constrain Lorentz violation in the photon sector by precise measurements of light bending at massive bodies. Note that the sensitivity does not largely depend on the model function used. This independence could be checked for further model functions, which can be regarded as an interesting future project. Anisotropic (nonbirefringent) case {#anisotropic-nonbirefringent-case} ---------------------------------- The anisotropic case of modified Maxwell theory exhibiting a single modified dispersion relation was discussed in . This particular case is characterized by a preferred spacelike direction (chosen to point along the positive $z$-axis) and one controlling coefficient. The refractive index was found in and it was expressed in terms of the angle $\vartheta$ enclosed between the propagation direction and the preferred axis. The possible trajectory of a light ray is parameterized by $\mathbf{r}(\phi)=r(\phi)\widehat{\mathbf{e}}_{\phi}$ such as for the isotropic case. The angle $\vartheta$ in the refractive index is then given by the scalar product of the tangent vector $\mathbf{t}$ and the preferred direction $\boldsymbol{\zeta}$ where it is sufficient to work in two spatial dimensions: $$\label{eq:anisotropic-angle} \cos\vartheta=\frac{\mathbf{t}\cdot \boldsymbol{\zeta}}{|\mathbf{t}|}=\frac{r(\phi)\cos\phi+\dot{r}(\phi)\sin\phi}{\sqrt{r(\phi)^2+\dot{r}(\phi)^2}}\,.$$ Note that for the anisotropic case angular momentum is not conserved and Bouguer’s formula loses its meaning. Hence there does not seem to be an alternative to solving the eikonal equation directly, which is carried out numerically for hypothetical values of $R_S$ and the controlling coefficient $\widetilde{\kappa}_{e-}^{11}$. The results are shown in . In contrast to the isotropic case, cf. , where the trajectory is not modified for a spacetime position independent $\widetilde{\kappa}_{\mathrm{tr}}$ this is not the case here. For the anisotropic sector the shape of the trajectory gets distorted where the final impact parameter decreases for $\widetilde{\kappa}_{e-}^{11}>0$. Physically this means that the ray loses angular momentum. An interesting future research project would be to perform a similar kind of phenomenological analysis as we did for the isotropic case. ![Solution of the eikonal equation in the $x$-$y$-plane (in dimensions of $d$) with $d/R_S=5$. The blue (plain) curve shows the solution for the Lorentz-invariant case, whereas the red (dashed) curve depicts the solution for the anisotropic case with $\widetilde{\kappa}_{e-}^{11}=0.65$. The massive body resides in the coordinate center.[]{data-label="fig:solution-eikonal-both-standard-and-anisotropic"}](solution-eikonal-angle-dependent-refractive-index.pdf) Modified energy-momentum conservation {#sec:modified-energy-momentum-conservation} ===================================== The phenomenology in the previous section was carried out in an explicitly Lorentz-violating framework, which is known to cause tensions in a gravitational background [@Kostelecky:2003fs]. The purpose of the current section is to investigate where exactly these problems occur in our classical description and how they can be interpreted from the point of view of an inhomogeneous medium. Therefore the energy-momentum tensor and its conservation law will be derived for the isotropic case. The (Belinfante-Rosenfeld) energy-momentum tensor follows from varying the corresponding Lagrangian with respect to the metric. The Finsler structure $F(\mathbf{u})|_{\circledcirc}^+$ of is the equivalent to a Lagrangian, since it appears as the integrand of the path length functional that is stationary for the trajectory travelled by the light ray. Instating an auxiliary metric tensor $\psi_{\mu\nu}$ leads to the following result: $$\label{eq:lagrangian-classical-light-ray-isotropic} F=n|\mathbf{u}|=\sqrt{\frac{1+\widetilde{\kappa}_{\mathrm{tr}}\,\psi_{\mu\nu}\xi^{\mu}\xi^{\nu}}{1-\widetilde{\kappa}_{\mathrm{tr}}\,\psi_{\rho\sigma}\xi^{\rho}\xi^{\sigma}}}\sqrt{-\psi_{ij}u^iu^j}\,.$$ Note that in Minkowski spacetime it holds that $\eta_{\mu\nu}\xi^{\mu}\xi^{\nu}=\xi^2=1$ and $-\eta_{ij}u^iu^j=\mathbf{u}^2$ where the minus sign in the latter term is due to the signature of the metric chosen. Variation has to be carried out for all independent degrees of freedom. A useful formula is $$\begin{aligned} \delta(A_{\mu}A^{\mu})&=\delta(\psi_{\mu\nu}A^{\mu}A^{\nu})=\psi_{\mu\nu}\delta A^{\mu}A^{\nu}+\psi_{\mu\nu}A^{\mu}\delta A^{\nu}+\delta \psi_{\mu\nu} A^{\mu}A^{\nu} \notag \\ &=2A_{\nu}\delta A^{\nu}+\delta \psi_{\mu\nu}A^{\mu}A^{\nu}\,,\end{aligned}$$ which states the variation of a scalar product of fields. Employing this rule, the variation of $F$ can then be computed as follows: $$\begin{aligned} \delta F&=(\delta n)\sqrt{-\psi_{ij}u^iu^j}+n\;\!\delta\left(\sqrt{-\psi_{ij}u^iu^j}\right) \notag \displaybreak[0]\\ &=\frac{\widetilde{\kappa}_{\mathrm{tr}}}{n(1-\widetilde{\kappa}_{\mathrm{tr}})^2}\delta\left(\psi_{\mu\nu}\xi^{\mu}\xi^{\nu}\right)\sqrt{-\psi_{ij}u^iu^j}+\frac{n}{2\sqrt{-\psi_{ij}u^iu^j}}\delta\left(-\psi_{ij}u^iu^j\right) \notag \displaybreak[0]\\ &=\frac{\widetilde{\kappa}_{\mathrm{tr}}}{n(1-\widetilde{\kappa}_{\mathrm{tr}})^2}\sqrt{-\psi_{ij}u^iu^j}\left(2\xi_{\nu}\delta\xi^{\nu}+\delta \psi_{\mu\nu}\xi^{\mu}\xi^{\nu}\right) \notag \displaybreak[0]\\ &\phantom{{}={}}-\frac{n}{2\sqrt{-\psi_{ij}u^iu^j}}\left(2u_j\delta u^j+\delta \psi_{ij}u^iu^j\right)\,.\end{aligned}$$ Now everything is available to obtain the energy-momentum tensor from $\delta F$ by considering all terms comprising a variation of the metric. An additional prefactor containing the metric has to be taken into account in the definition. However we are interested in the covariant conservation law of $T^{\mu\nu}$ for Minkowski spacetime, i.e., for a spacetime-position dependent refractive index without an additional gravitational field. In this case $\psi_{\mu\nu}=\eta_{\mu\nu}$ whereby $$\begin{aligned} T^{\mu\nu}&\equiv\left.\frac{2}{\sqrt{|\psi|}}\frac{\delta (\sqrt{|\psi|}F)}{\delta \psi_{\mu\nu}}\right|_{\psi_{\mu\nu}=\eta_{\mu\nu}}=\frac{2\widetilde{\kappa}_{\mathrm{tr}}}{n(1-\widetilde{\kappa}_{\mathrm{tr}})^2}\sqrt{u_iu^i}\xi^{\mu}\xi^{\nu}-\frac{n}{\sqrt{u_iu^i}}\widetilde{u}^{\mu}\widetilde{u}^{\nu} \notag \\ &=n\sqrt{u_iu^i}\left[\frac{1}{2}\left(n^2-\frac{1}{n^2}\right)\xi^{\mu}\xi^{\nu}-\frac{\widetilde{u}^{\mu}\widetilde{u}^{\nu}}{u_iu^i}\right]\,.\end{aligned}$$ Here $\psi\equiv\det(\psi_{\mu\nu})$ and $(\widetilde{u}^{\mu})\equiv (0,\mathbf{u})^T$, i.e., $\widetilde{u}^{\mu}$ involves the spatial velocity and its zeroth component vanishes. Upon inspection of the latter result we see that the 00-component of $T^{\mu\nu}$ is made up by the preferred timelike spacetime direction $\xi^{\mu}$ and it vanishes for $n=1$, i.e., in a Lorentz-invariant vacuum. The spatial part solely comprises products of three-velocity components and the mixed components vanish. Now the partial derivative of the energy-momentum tensor in Minkowski spacetime leads to: $$\begin{aligned} \label{eq:conservation-energy-momentum} \partial_{\mu}T^{\mu\nu}&=(\partial_{\mu}n)\sqrt{\mathbf{u}^2}\left[\frac{1}{2}\left(n^2-\frac{1}{n^2}\right)\xi^{\mu}\xi^{\nu}-\frac{\widetilde{u}^{\mu}\widetilde{u}^{\nu}}{\mathbf{u}^2}\right]+n\sqrt{\mathbf{u}^2}\partial_{\mu}\left[\frac{1}{2}\left(n^2-\frac{1}{n^2}\right)\right]\xi^{\mu}\xi^{\nu} \notag \\ &=\frac{T^{\mu\nu}}{n}\partial_{\mu}n+\left(n^2+\frac{1}{n^2}\right)\sqrt{\mathbf{u}^2}\,\xi^{\mu}\xi^{\nu}(\partial_{\mu}n) \notag \\ &=\sqrt{\mathbf{u}^2}\left[\frac{1}{2}\left(3n^2+\frac{1}{n^2}\right)\xi^{\mu}\xi^{\nu}-\frac{\widetilde{u}^{\mu}\widetilde{u}^{\nu}}{\mathbf{u}^2}\right](\partial_{\mu}n)\,.\end{aligned}$$ An interesting observation is that the timelike contribution can be expressed in terms of the metric $\widetilde{g}_{\mu\nu}$ appearing in : $$\begin{aligned} (\widetilde{g}^2)^{\mu}_{\phantom{\mu}\mu}&=3n^2+\frac{1}{n^2}\,, \\[2ex] \label{eq:four-dimensional-isotropic-metric} \widetilde{g}_{\mu\nu}(r)&\equiv \mathrm{diag}\left(\frac{1}{n(r)},-n(r),-n(r),-n(r)\right)_{\mu\nu}\,.\end{aligned}$$ Note that $\widetilde{g}_{\mu\nu}$ is not associated to a gravity field but only to a nonconstant refractive index. The result obtained in describes the conservation of energy and momentum of a light ray. Its properties are in order. First, it vanishes for a constant refractive index, i.e., energy and momentum of the ray are conserved in a homogeneous medium, in the Lorentz-invariant vacuum, and a Lorentz-violating vacuum with a constant controlling coefficient. Second, in an inhomogeneous medium or in a Lorentz-violating vacuum with spacetime-dependent controlling coefficient the energy-momentum tensor is not conserved, since the partial derivative of the refractive index does not vanish in this case. As long as the refractive index is not time-dependent, $\partial_0n=0$, which is why $\partial_{\mu}T^{\mu 0}=0$. Since the case under consideration is isotropic, the controlling coefficient and the refractive index, respectively, can only depend on the radial coordinate: $\widetilde{\kappa}_{\mathrm{tr}}=\widetilde{\kappa}_{\mathrm{tr}}(r)$, $n=n(r)$. Hence $\partial_{\mu}n$ has a nonvanishing component along the radial basis vector only, i.e., $\partial_rn\neq 0$ and $\partial_{\theta}n=\partial_{\phi}n=0$. Decomposing the spatial velocity into a radial part $u^r$ and transverse components $u^{\theta}$, $u^{\phi}$, $$\mathbf{u}=u^r\mathbf{e}_r+u^{\theta}\mathbf{e}_{\theta}+u^{\phi}\mathbf{e}_{\phi}\,,$$ the spatial part of the conservation law reads as $$\label{eq:modified-energy-momentum-conservation-isotropic} \partial_{\mu}T^{\mu i}=-\frac{1}{\sqrt{\mathbf{u}^2}}(\boldsymbol{\nabla}n\cdot\mathbf{u})u^i=-\sqrt{\mathbf{u}^2}(\boldsymbol{\nabla}n\cdot\widehat{\mathbf{u}})\widehat{u}^i =-\sqrt{\mathbf{u}^2}(\partial_rn)\widehat{u}^r\widehat{u}^i\,,\quad \widehat{\mathbf{u}}=\frac{\mathbf{u}}{|\mathbf{u}|}\,.$$ Several observations can be made upon inspecting the result. For a constant refractive index the right-hand side of the latter equation is zero, which means that the spatial part of the energy-momentum conservation law is valid as well in this case. For $\partial_rn\neq 0$ it even holds when the radial velocity component vanishes: $u^r=0$. This is a special situation that can occur for a light ray in an inhomogeneous, isotropic medium whose refractive index has a particular $r$-dependence and when the ray is emitted tangentially to a circle with its center lying in the coordinate origin (cf. [@Evans:1985] for a beautiful paper on geometric-ray optics and its implications for certain optical systems). The trajectory of the ray is then a circle where the refractive index is constant. The magnitude of the three-momentum vector does not change, but only its direction. So momentum is not exchanged between the light ray and the medium, because any momentum transfer would change the magnitude of the momentum vector. For any other case with nonzero $\partial_rn$ momentum has to be exchanged, which is why $T^{\mu\nu}$ of the ray cannot be conserved. The net term obtained above points in the direction $\widehat{\mathbf{u}}$ of the ray at the point considered. However the total energy-momentum tensor with $T^{\mu\nu}_{\mathrm{med}}$ of the medium included is expected to be conserved, because any momentum change of the light ray will cause a momentum change of the medium itself. In General Relativity local diffeomorphism invariance is tightly linked to energy-momentum conservation. In [@Kostelecky:2003fs] it was shown that explicit Lorentz violation in gravity leads to a loss of diffeomorphism invariance, which then causes the energy-momentum tensor to be no longer covariantly conserved. Note that in the latter reference the energy-momentum tensor $T_e^{\mu\nu}$ is considered that follows by varying the Lagrangian with respect to the vierbein instead of with respect to the metric tensor. It is different from the Belinfante-Rosenfeld energy-momentum tensor considered here even in case there is no Lorentz violation [@Belinfante:1940]. The covariant derivative of $T_e^{\mu\nu}$ in [@Kostelecky:2003fs] involves the covariant derivative of the Lorentz-violating controlling coefficients, i.e., a term of the structure $J^xD_{\nu}k_x$. Here $k_x$ is a generic controlling coefficient with a particular Lorentz index structure $x$ contracted with an appropriate operator $J^x$. For general curved manifolds there is no spacetime-position dependent function satisfying $D_{\nu}k_x=0$, but only for parallelizable manifolds such as the circle $S^1$ or the two-torus $T^2=S^1\times S^1$. In four dimensions such manifolds are rare, though, and they do not seem to be of particular interest in the context of General Relativity. Note that the conservation law without gravitational fields is given by $\partial_{\mu}(\Theta_c)^{\mu\nu}=J^x\partial^{\nu}k_x$ with the canonical energy-momentum tensor $(\Theta_c)^{\mu\nu}$ [@Kostelecky:2003fs]. Therefore if the controlling coefficient $k_x$ is dependent on spacetime position the conservation law is modified even in flat spacetime. From a physical perspective this is not surprising, since such a controlling coefficient implies that the vacuum behaves like an effective, inhomogeneous medium. In general the magnitude of the three-momentum of a light ray is not conserved as was argued above. Therefore momentum has to be exchanged between the ray and the medium. When considering explicit Lorentz violation the effective medium is considered to be nondynamical, which is why it can neither absorb nor deliver momentum to the light ray. Interestingly the situation is different when spontaneous violations of diffeomorphism invariance and local Lorentz symmetry are considered. In these cases the ground state violates these symmetries dynamically by an emergent vacuum expectation value of a vector or tensor field in a potential [@Kostelecky:2003fs; @Kostelecky:1988zi; @Kostelecky:1989jp; @Kostelecky:1989jw; @Bailey:2006fd; @Bluhm:2008yt; @Hernaski:2014jsa; @Bluhm:2014oua]. Such models have in common that they involve massless (Nambu-Goldstone) modes where the latter appear when any global, continuous symmetry is broken spontaneously. When the symmetry is local there can be an additional Higgs-type mechanism absorbing the massless modes to produce massive gauge fields. Since for spontaneous Lorentz violation the dynamics of the Lorentz-violating background field is taken into account, the energy-momentum conservation law is restored in these theories. In the corresponding equation there is no contribution $J^x\partial^{\nu}k_x$. From the perspective of an inhomogeneous medium translational and rotational symmetry are violated spontaneously by the atomic lattice. The Nambu-Goldstone modes (gapless excitations) linked to the spontaneous violation of these symmetries are the two transverse and the longitudinal types of phonons.[^3] Since the medium is now dynamical, it can absorb momentum from the ray upon producing phonons. Hence the conservation law for the light ray remains valid in this case. Properties of the isotropic Finsler space {#sec:properties-isotropic-finsler} ========================================= In the previous section the modified conservation law for the Belinfante-Rosenfeld energy-momentum tensor for a light ray in an isotropic, inhomogeneous medium was considered and discussed. It was found that the energy-momentum tensor is not conserved, since the nontrivial medium is nondynamical corresponding to a background violating Lorentz symmetry explicitly. This issue persists even in Minkowski spacetime, since in inhomogeneous media light rays behave similarly to when they propagate in gravitational backgrounds. The most prominent example for a common effect is light bending. The conservation law of the energy-momentum tensor for a medium with spherically symmetric refractive index, , involves both the first derivative of the refractive index and the propagation direction of the ray at a given point. Recall that in gravitational theories with explicit Lorentz violation nonconservation of the energy-momentum tensor clashes with the Bianchi identities of Riemannian geometry [@Kostelecky:2003fs]. Although we work with an effective Lorentz-violating theory for light rays based on a nontrivial refractive index, this issue can be encountered here as well. The purpose of the current section is to figure out whether explicit (isotropic) Lorentz violation can be considered in a weak gravity field in the framework of Finsler geometry such that no inconsistencies arise. As a basis we use the spacetime metric of , which was shown to be closely linked to the isotropic case. The properties of the latter metric shall be studied from a Finslerian point of view where we use the conventions of [@Bao:2000] for all geometrical quantities. The latter are treated based on the indefinite signature of the metric in . As a starting point an appropriate Finsler structure has to be constructed whose derived metric should correspond to . This works for the following choice: $$\label{eq:finsler-structure-isotropic-refractive-index} F(r,\mathbf{y})=\sqrt{\mathbf{y}^2}\,,\quad \mathbf{y}^2=\frac{1}{n}(y^t)^2-n(y^r)^2-n\left[(y^{\theta})^2+(y^{\phi})^2\sin^2\theta\right]r^2\,,$$ where the refractive index solely has a radial dependence. In what follows we write $n(r)=n$ for brevity, i.e., the argument of the refractive index will be omitted. The vector $\mathbf{y}\in TM$ is expressed in spherical polar coordinates as $\mathbf{y}=y^t\mathbf{e}_t+y^r\mathbf{e}_r+y^{\theta}\mathbf{e}_{\theta}+y^{\phi}\mathbf{e}_{\phi}$ with suitable basis vectors. The spatial part of the Finsler structure is written in terms of spherical polar coordinates. The corresponding Finsler metric is then computed according to the usual definition and it corresponds to the result of (with the spatial part transformed to spherical coordinates): $$\label{eq:finsler-metric-isotropic} g_{\mu\nu}\equiv\frac{1}{2}\frac{\partial^2F^2}{\partial y^{\mu}\partial y^{\nu}}=\mathrm{diag}\left(\frac{1}{n},-n,-nr^2,-nr^2\sin^2\theta\right)_{\mu\nu}\,.$$ The inverse metric simply reads as $$\label{eq:finsler-metric-isotropic-inverse} g^{\mu\nu}=\mathrm{diag}\left(n,-\frac{1}{n},-\frac{1}{nr^2},-\frac{1}{nr^2\sin^2\theta}\right)^{\mu\nu}\,.$$ Since the metric does not depend on $\mathbf{y}$, the Cartan connection [@Bao:2000] vanishes: $$\label{eq:cartan-torsion-isotropic} A_{\mu\nu\varrho}\equiv \frac{F}{2}\frac{\partial g_{\mu\nu}}{\partial y^{\varrho}}=\frac{F}{4}\frac{\partial^3F^2}{\partial y^{\mu}\partial y^{\nu}\partial y^{\varrho}}\,,$$ Therefore according to Deicke’s theorem [@Deicke:1953] the Finsler space considered is Riemannian. Now the base has been set up to study the geometry of the space defined by . The first step is the obtain the coefficients of the affine connection (Christoffel symbols of second kind) that are defined in analogy to the Christoffel symbols in Riemannian geometry: $$\label{eq:christoffel-symbols} \gamma^{\mu}_{\phantom{\mu}\nu\rho}=\frac{1}{2}g^{\;\!\mu\alpha}\left(\frac{\partial g_{\alpha\nu}}{\partial x^{\rho}}-\frac{\partial g_{\nu\rho}}{\partial x^{\alpha}}+\frac{\partial g_{\rho\alpha}}{\partial x^{\nu}}\right)\,.$$ Note that summation over equal indices is understood based on Einstein’s convention. The nonzero contributions read as follows: $$\begin{aligned} \gamma^t_{\phantom{t}tr}&=-\frac{n'}{2n}\,,\quad \gamma^r_{\phantom{r}tt}=-\frac{n'}{2n^3}\,,\quad \gamma^r_{\phantom{r}rr}=\frac{n'}{2n}\,,\quad \gamma^r_{\phantom{r}\theta\theta}=-\frac{r(2n+rn')}{2n}\,, \\[2ex] \gamma^r_{\phantom{r}\phi\phi}&=-\frac{r(2n+rn')}{2n}\sin^2\theta\,,\quad \gamma^{\theta}_{\phantom{\theta}r\theta}=\frac{1}{r}+\frac{n'}{2n}\,,\quad \gamma^{\theta}_{\phantom{\theta}\phi\phi}=-\sin\theta\cos\theta\,, \\[2ex] \gamma^{\phi}_{\phantom{\phi}r\phi}&=\frac{1}{r}+\frac{n'}{2n}\,,\quad \gamma^{\phi}_{\phantom{\phi}\theta\phi}=\cot\theta\,.\end{aligned}$$ Since torsion is assumed to vanish, the Christoffel symbols are symmetric in the latter two indices. The connection coefficients with at least one index equal to the radial coordinate $r$ involve the first derivative of the refractive index. Furthermore they do not involve the angle $\phi$ as expected for spherically symmetric metrics. As a next step the geodesic spray coefficients are needed: $G^{\mu}\equiv\gamma^{\mu}_{\phantom{\mu}\nu\varrho}y^{\nu}y^{\varrho}$. The latter appear in the geodesic equations in Finsler geometry: $$\begin{aligned} G^t&=-y^ty^r\frac{n'}{n}\,, \\[2ex] G^r&=-(y^t)^2\frac{n'}{2n^3}+\frac{1}{2n}\left\{(y^r)^2n'-r(2n+rn')[(y^{\theta})^2+(y^{\phi})^2\sin^2\theta]\right\}\,, \\[2ex] G^{\theta}&=y^ry^{\theta}\left(\frac{2}{r}+\frac{n'}{n}\right)-(y^{\phi})^2\sin\theta\cos\theta\,,\quad G^{\phi}=y^{\phi}\left[2y^{\theta}\cot\theta+y^r\left(\frac{2}{r}+\frac{n'}{n}\right)\right]\,.\end{aligned}$$ The geodesic spray coefficients can be used to define the nonlinear connection [@Bao:2000] on $TM\setminus \{0\}$: $$N^{\mu}_{\phantom{\mu}\nu}\equiv\frac{1}{2}\frac{\partial G^{\mu}}{\partial y^{\nu}}\,.$$ The reasons for introducing these connection coefficients is as follows. On the one hand the basis vectors $\partial/\partial x^{\nu}$ and $\partial/\partial y^{\nu}$ are unsuitable to be chosen as a local basis of $TTM$, since the $\partial/\partial x^{\nu}$ transform in a complicated way. On the other hand if $\{\mathrm{d}x^{\mu},\mathrm{d}y^{\mu}\}$ is chosen as a local basis of the cotangent bundle $T^{*}TM$ the transformation properties of $\mathrm{d}y^{\mu}$ are involved. To have the desired transformation properties for the basis of the tangent and the cotangent bundle of $TM\setminus \{0\}$ the following basis vectors can be introduced using the nonlinear connection: $$\begin{aligned} \left\{\frac{\delta}{\delta x^{\nu}},F\frac{\partial}{\partial y^{\nu}}\right\}\,,&\quad \frac{\delta}{\delta x^{\nu}}\equiv \frac{\partial}{\partial x^{\nu}}-N^{\mu}_{\phantom{\mu}\nu}\frac{\partial}{\partial y^{\mu}}\,, \\[2ex] \left\{\mathrm{d}x^{\mu},\frac{\delta y^{\mu}}{F}\right\}\,,&\quad \delta y^{\mu}\equiv \mathrm{d}y^{\mu}+N^{\mu}_{\phantom{\mu}\nu}\mathrm{d}x^{\nu}\,.\end{aligned}$$ For the particular case studied here the nonlinear connection coefficients can be comprised in a $(3\times 3)$ matrix that reads as $$(N^{\mu}_{\phantom{\mu}\nu})=\frac{1}{2}\begin{pmatrix} -y^rn'/n & -y^tn'/n & 0 & 0 \\ -y^tn'/n^3 & y^rn'/n & -y^{\theta}r(2n+rn')/n & -y^{\phi}r\sin^2\theta(2n+rn')/n \\ 0 & y^{\theta}(2/r+n'/n) & y^r(2/r+n'/n) & -y^{\phi}\sin(2\theta) \\ 0 & y^{\phi}(2/r+n'/n) & 2y^{\phi}\cot\theta & y^r(2/r+n'/n)+2y^{\theta}\cot\theta \\ \end{pmatrix}\,.$$ To compute directional derivatives of tensor fields on Finsler manifolds, a further connection has to be found to define a covariant derivative. It was shown that the pulled-back bundle $\pi^{*}TM$ has a linear connection associated to it, which is called the Chern connection $\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}$. Explicitly it can be obtained from the Finsler metric using the nonlinear connection $N^{\mu}_{\phantom{\mu}\nu}$: $$\begin{aligned} \Gamma^{\mu}_{\phantom{\mu}\nu\varrho}&=\frac{1}{2}g^{\mu\alpha}\left(\frac{\delta g_{\alpha\nu}}{\delta x^{\varrho}}-\frac{\delta g_{\nu\varrho}}{\delta x^{\alpha}}+\frac{\delta g_{\varrho\alpha}}{\delta x^{\nu}}\right) \notag \\ &=\frac{1}{2}g^{\mu\alpha}\left(\frac{\partial g_{\alpha\nu}}{\partial x^{\varrho}}-N^{\beta}_{\phantom{\beta}\varrho}\frac{\partial g_{\alpha\nu}}{\partial y^{\beta}}-\left[\frac{\partial g_{\nu\varrho}}{\partial x^{\alpha}}-N^{\beta}_{\phantom{\beta}\alpha}\frac{\partial g_{\nu\varrho}}{\partial y^{\beta}}\right]+\frac{\partial g_{\varrho\alpha}}{\partial x^{\nu}}-N^{\beta}_{\phantom{\beta}\nu}\frac{\partial g_{\varrho\alpha}}{\partial y^{\beta}}\right)\,.\end{aligned}$$ The Chern connection is unique and formally it has the same index structure as the formal Christoffel symbols. The difference to the latter is that the derivative $\delta/\delta x^{\mu}$ is used instead of the ordinary partial derivative $\partial/\partial x^{\mu}$. However in the particular case studied here, $\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}=\gamma^{\mu}_{\phantom{\mu}\nu\varrho}$, since the Finsler metric $g_{\mu\nu}$ does not depend on the components of $\mathbf{y}$. Finally the Chern connection is needed to define a Finslerian version of the Riemann curvature tensor: $$\begin{aligned} \label{eq:finsler-curvature-tensor} R_{\nu\phantom{\mu}\varrho\sigma}^{\phantom{\nu}\mu}&=\frac{\delta\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}}{\delta x^{\varrho}}-\frac{\delta\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}}{\delta x^{\sigma}}+\Gamma^{\mu}_{\phantom{\mu}\alpha\varrho}\Gamma^{\alpha}_{\phantom{\alpha}\nu\sigma}-\Gamma^{\mu}_{\phantom{\mu}\alpha\sigma}\Gamma^{\alpha}_{\phantom{\alpha}\nu\varrho} \notag \\ &=\frac{\partial\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}}{\partial x^{\varrho}}-N^{\beta}_{\phantom{\beta}\varrho}\frac{\partial\Gamma^{\mu}_{\phantom{\mu}\nu\sigma}}{\partial y^{\beta}}-\left(\frac{\partial\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}}{\partial x^{\sigma}}-N^{\beta}_{\phantom{\beta}\sigma}\frac{\partial\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}}{\partial y^{\beta}}\right)+\Gamma^{\mu}_{\phantom{\mu}\alpha\varrho}\Gamma^{\alpha}_{\phantom{\alpha}\nu\sigma}-\Gamma^{\mu}_{\phantom{\mu}\alpha\sigma}\Gamma^{\alpha}_{\phantom{\alpha}\nu\varrho}\,.\end{aligned}$$ Since the Chern connection coefficients correspond to the formal Christoffel symbols and the latter are independent of $y^{\mu}$, the curvature components correspond to the Riemannian ones. They involve an additional derivative of the Christoffel symbols, which is why they comprise second derivatives of the refractive index. Explicitly the independent curvature tensor components are stated as follows: $$\begin{aligned} R_{t\phantom{r}tr}^{\phantom{t}r}&=\frac{nn''-2n'^2}{2n^4}\,,\quad R_{t\phantom{\theta}t\theta}^{\phantom{t}\theta}=\frac{n'(2n+rn')}{4rn^4}=R_{t\phantom{\phi}t\phi}^{\phantom{t}\phi}\,, \\[2ex] R_{r\phantom{t}tr}^{\phantom{r}t}&=\frac{nn''-2n'^2}{2n^2}\,,\quad R_{\theta\phantom{t}t\theta}^{\phantom{\theta}t}=\frac{rn'(2n+rn')}{4n^2}\,,\quad R_{\theta\phantom{\phi}\theta\phi}^{\phantom{\theta}\phi}=\frac{rn'(4n+rn')}{4n^2}\,, \\[2ex] R_{\phi\phantom{t}t\phi}^{\phantom{\phi}t}&=\frac{rn'(2n+rn')}{4n^2}\sin^2\theta\,,\quad R_{\phi\phantom{\theta}\theta\phi}^{\phantom{\phi}\theta}=-\frac{rn'(4n+rn')}{4n^2}\sin^2\theta\,.\end{aligned}$$ The components related by symmetries are omitted. Since the Finsler structure of is Riemannian according to Deicke’s theorem, we will first use the Riemannian definitions of the Ricci tensor $\mathcal{Ric}_{\mu\nu}\equiv R_{\mu\phantom{\alpha}\alpha\nu}^{\phantom{\mu}\alpha}$ and the curvature scalar (Ricci scalar) $\mathcal{Ric}$. These are denoted by calligraphic letters and they follow from suitable contractions of the Riemann curvature tensor. The Ricci tensor components with equal indices deliver nonzero contributions only: $$\begin{aligned} \mathcal{Ric}_{tt}&=\frac{1}{2rn^4}\left[rn'^2-n(2n'+rn'')\right]\,,\quad \mathcal{Ric}_{rr}=-\frac{1}{2rn}(2n'+rn'')\,, \\[2ex] \mathcal{Ric}_{\theta\theta}&=\frac{r}{2n^2}\left[rn'^2-n(2n'+rn'')\right]\,,\quad \mathcal{Ric}_{\phi\phi}=\frac{r}{2n^2}\left[rn'^2-n(2n'+rn'')\right]\sin^2\theta\,, \\[2ex] \label{eq:ricci-scalar-conventional} \mathcal{Ric}&\equiv \mathcal{Ric}^{\mu}_{\phantom{\mu}\mu}=g^{\mu\nu}\mathcal{Ric}_{\mu\nu}=\frac{1}{2rn^3}\left[2n(2n'+rn'')-rn'^2\right]\,.\end{aligned}$$ In Riemannian geometry the curvature tensor obeys the first and the second Bianchi identities. Especially the second one, $$\begin{aligned} 0&\equiv D_{\eta}R_{\lambda\phantom{\mu}\nu\kappa}^{\phantom{\lambda}\mu}+D_{\kappa}R_{\lambda\phantom{\mu}\eta\nu}^{\phantom{\lambda}\mu}+D_{\nu}R_{\lambda\phantom{\mu}\kappa\eta}^{\phantom{\lambda}\mu}\,, \\[2ex] D_{\lambda}R_{\mu\phantom{\nu}\rho\sigma}^{\phantom{\mu}\nu}&=\partial_{\lambda}R_{\mu\phantom{\nu}\rho\sigma}^{\phantom{\mu}\nu}-\Gamma^{\alpha}_{\phantom{\alpha}\mu\lambda}R_{\alpha\phantom{\nu}\rho\sigma}^{\phantom{\alpha}\nu}+\Gamma^{\nu}_{\phantom{\nu}\alpha\lambda}R_{\mu\phantom{\alpha}\rho\sigma}^{\phantom{\mu}\alpha}-\Gamma^{\alpha}_{\phantom{\alpha}\rho\lambda}R_{\mu\phantom{\nu}\alpha\sigma}^{\phantom{\mu}\nu}-\Gamma^{\alpha}_{\phantom{\alpha}\sigma\lambda}R_{\mu\phantom{\nu}\rho\alpha}^{\phantom{\mu}\nu}\,,\end{aligned}$$ is important in the context of General Relativity, because it leads to the statement that the Einstein tensor $G^{\mu\nu}$ is covariantly constant: $$\begin{aligned} D_{\mu}G^{\mu}_{\phantom{\mu}\nu}=\partial_{\mu}G^{\mu}_{\phantom{\mu}\nu}+\Gamma^{\mu}_{\phantom{\mu}\alpha\mu}G^{\alpha}_{\phantom{\alpha}\nu}-\Gamma^{\alpha}_{\phantom{\alpha}\nu\mu}G^{\mu}_{\phantom{\mu}\alpha}\equiv 0\,,\quad G^{\mu\nu}\equiv \mathcal{Ric}^{\mu\nu}-\frac{\mathcal{Ric}}{2}g^{\mu\nu}\,,\end{aligned}$$ which was checked to be valid for the particular metric $g_{\mu\nu}$ of . This identity is the reason why explicit Lorentz violation is incompatible with Riemannian geometry. Due to the Einstein equations it forces the energy-momentum tensor to be covariantly conserved as well, which does not necessarily hold when there is a spacetime-dependent background. At this point it is reasonable to wonder how Finsler geometry can help us to solve that problem. For the isotropic metric considered the identity $D_{\mu}G^{\mu}_{\phantom{\mu}\nu}\equiv 0$ is inherited from the Riemannian to the Finslerian framework, since the Finsler metric of does not comprise any dependence on $y^{\mu}$. Assuming that Finsler geometry provides the necessary tools to circumvent the no-go theorem of [@Kostelecky:2003fs] in a general explicitly Lorentz-violating setting, then it should also work for the special isotropic case studied here. One possible approach (there may be others) might be to consider a suitable equivalent of the Einstein equations in Finsler geometry. Such an equivalent can be based on an alternative definition of the Einstein tensor $G^{\mu\nu}$ constructed from curvature-related tensors in the Finsler framework. These objects will be introduced in what follows. The first is obtained from the curvature tensor by contracting the latter with two vectors $y^{\mu}/F$ according to $$\label{eq:predecessor-flag-curvature} R^{\mu}_{\phantom{\mu}\varrho}\equiv\frac{y^{\nu}}{F}R_{\nu\phantom{\mu}\varrho\sigma}^{\phantom{\nu}\mu}\frac{y^{\sigma}}{F}\,.$$ Note that this construction does not correspond to the Ricci tensor of Riemannian geometry. In particular it is sometimes referred to as the predecessor of flag curvature, which is a generalization of sectional curvature in Finsler geometry. For the special case here $R^{\mu}_{\phantom{\mu}\varrho}$ are the components of a $(4\times 4)$ matrix. The trace of this matrix is taken to obtain the generalization of the Ricci scalar in Finsler geometry: $\mathit{Ric}\equiv R^{\varrho}_{\phantom{\varrho}\varrho}$. Since the explicit expressions for $R^{\mu}_{\phantom{\mu}\varrho}$ and $\textit{Ric}$ are complicated and not illuminating, they will not be stated explicitly. The flag curvature in Finsler geometry is computed similarly to the sectional curvature in Riemannian geometry. The latter is defined in a tangent space at a point $x$ of the manifold where two arbitrary, linearly independent directions are needed for its computation. The resulting quantity only depends on the plane considered, but not on the particular choice of the directions. The flag curvature in Finsler geometry carries the same spirit where one direction is chosen to correspond to $\mathbf{y}$ and the other one, say $\mathbf{L}$, is supposed to be orthogonal to $\mathbf{y}$. These vectors are then suitably contracted with the curvature tensor of . Note that $\mathbf{y}$ and the vector orthogonal to it can be considered to span a flag where $\mathbf{y}$ is assumed to point along the flag pole. This explains the name for the curvature. For an $n$-dimensional Finsler manifold $R$ is the sum of $n-1$ flag curvatures. It only depends on $r$ and $\mathbf{y}$, but not on the direction $\mathbf{L}$ chosen orthogonal to $\mathbf{y}$. Although $R_{\mu\varrho}$ of is not understood to be the generalization of the Ricci tensor in Finsler geometry, it is still possible to define the latter. The definition (cf. Eq. (7.6.4) in [@Bao:2000]) involves both the Finsler structure $F$ and the Finslerian version of the Ricci scalar $\textit{Ric}$: $$\label{eq:ricci-tensor-from-predecessor} \mathit{Ric}_{\mu\nu}\equiv \frac{1}{2}\frac{\partial^2(F^2\textit{Ric})}{\partial y^{\mu}\partial y^{\nu}}\,.$$ For Finsler metrics that are Riemannian, i.e., for the isotropic metric considered in it also holds that $\mathit{Ric}_{\mu\nu}=R_{\mu\phantom{\alpha}\alpha\nu}^{\phantom{\mu}\alpha}$. Hence in our case the Finslerian definition of the Ricci tensor corresponds to the Riemannian expression, computing an appropriate trace of the curvature tensor. The expression of can be used to obtain the Ricci scalar in Finsler geometry by contracting the Ricci tensor with two vectors $y^{\mu}/F$ (cf. (7.6.5) in [@Bao:2000]): $$\label{eq:ricci-scalar-finsler} \mathit{Ric}\equiv \mathit{Ric}_{\mu\nu}\frac{y^{\mu}}{F}\frac{y^{\nu}}{F}\,.$$ It can be shown in general that the latter corresponds to $R^{\varrho}_{\phantom{\varrho}\varrho}$ that is obtained from tracing . This object is distinguished from the Ricci scalar $\mathcal{Ric}$ in a Riemannian setting, which follows from tracing the Ricci curvature tensor $\mathcal{Ric}_{\mu\nu}$, cf. . Note that the quantity of is the direct Finslerian equivalent of the Ricci scalar. Since the Finsler metric considered is Riemannian, $\mathcal{Ric}$ only involves dependences on $r$, whereas $\textit{Ric}$ depends on $y^{\mu}$ as well. In general and especially here $\mathcal{Ric}\neq \textit{Ric}$. At this stage there are several possibilities of defining the Einstein tensor $G_{\mu\nu}$ in a Finsler framework using different combinations of $\mathcal{Ric}_{\mu\nu}$, $\mathcal{Ric}$, $R_{\mu\nu}$, $\mathit{Ric}_{\mu\nu}$, and $\mathit{Ric}$. The following have been tried: \[eq:propositions-einstein-tensor\] $$\begin{aligned} (G^{\mu}_{\phantom{\mu}\nu})^{(1)}&\equiv \mathcal{Ric}^{\mu}_{\phantom{\mu}\nu}-\frac{1}{2}\delta^{\mu}_{\phantom{\mu}\nu}\textit{Ric}\,, \\[2ex] (G^{\mu}_{\phantom{\mu}\nu})^{(2)}&\equiv R^{\mu}_{\phantom{\mu}\nu}-\frac{1}{2}\delta^{\mu}_{\phantom{\mu}\nu}\textit{Ric}\,, \\[2ex] (G^{\mu}_{\phantom{\mu}\nu})^{(3)}&\equiv R^{\mu}_{\phantom{\mu}\nu}-\frac{1}{2}\delta^{\mu}_{\phantom{\mu}\nu}\mathcal{Ric}\,.\end{aligned}$$ A reasonable test of whether one of these choices is suitable, requires computing their covariant derivatives, i.e., $D_{\mu}(G^{\mu}_{\phantom{\mu}\nu})^{(i)}$ for $i=1\dots 3$. The wishful result would be a nonvanishing covariant derivative bearing resemblance to the modified covariant conservation law of the energy-momentum tensor in . This makes sense when we assume that the modified Einstein tensor (in a Finslerian framework) is linked to the energy-momentum tensor in an explicitly Lorentz-violating theory. The corresponding covariant derivative to be used involves both the nonminimal connection $N^{\mu}_{\phantom{\mu}\nu}$ and the Chern connection $\Gamma^{\mu}_{\phantom{\mu}\nu\varrho}$ being equal to the Christoffel symbols $\gamma^{\mu}_{\phantom{\mu}\nu\varrho}$ in this case: $$\label{eq:covariant-derivative-finsler-einstein-tensor} D_{\mu}(G^{\mu}_{\phantom{\mu}\nu})^{(i)}=\frac{\partial(G^{\mu}_{\phantom{\mu}\nu})^{(i)}}{\partial x^{\mu}}-N^{\beta}_{\phantom{\beta}\mu}\frac{(G^{\mu}_{\phantom{\mu}\nu})^{(i)}}{\partial y^{\beta}}+\Gamma^{\mu}_{\phantom{\mu}\alpha\mu}(G^{\alpha}_{\phantom{\alpha}\nu})^{(i)}-\Gamma^{\alpha}_{\phantom{\alpha}\nu\mu}(G^{\mu}_{\phantom{\mu}\alpha})^{(i)}\,.$$ Starting from the Finsler metric of there have been up to three derivatives with respect to the coordinates involved, which is why in the third derivative of the refractive index appears in general. The more of the higher derivatives of a Taylor expansion of $n(r)$ are taken into account, the smaller are the structures in changes of $n(r)$ to be resolved. Therefore relying on the geometric-optics approximation it is reasonable to consider only the first-order change of $n(r)$ incorporated in its first derivative and to neglect the higher-order derivatives, which describe small-scale changes of $n(r)$. Within this approximation it makes sense to set $n(r)=1$, since modifications lead to higher-order contributions. Furthermore the Finsler structure that the isotropic case was identified with is three-dimensional, cf. , and it involves spatial velocity components only. Hence $y^t$ can be considered as auxiliary and will be set to zero at the end. With this physical input the covariant derivative of each Einstein tensor proposed in can be computed. The final result for the third possibility looks rather promising: $$D_{\mu}(G^{\mu}_{\phantom{\mu}\nu})^{(3)}|_{y^t=0}=\frac{1}{(y^r)^2+r^2[(y^{\theta})^2+(y^{\phi})^2\sin^2\theta]}\frac{n'}{r^2}\begin{pmatrix} 0 \\ (y^r)^2 \\ y^ry^{\theta}r^2 \\ y^ry^{\phi}r^2\sin^2\theta \\ \end{pmatrix}_{\nu}+\dots\,,$$ where terms of $\mathcal{O}(n'',n''',n'^2,n'^3)$ have been neglected. Using the inverse metric $g^{\mu\nu}$ of the second index can be raised. Besides we identify the spatial components of $\mathbf{y}$ with the spatial components of the physical velocity, i.e., $y^r=u^r$, $y^{\theta}=u^{\theta}$, and $y^{\phi}=u^{\phi}$ where the spatial flat metric in spherical polar coordinates is given by $(r_{ij})=\mathrm{diag}(1,r^2,r^2\sin^2\theta)$. This leads to the final result $$\begin{aligned} D_{\mu}(G^{\mu\nu})^{(3)}|_{y^t=0}&=-\frac{n'}{\mathbf{u}^2r^2}\begin{pmatrix} 0 \\ (u^r)^2 \\ u^ru^{\theta} \\ u^ru^{\phi} \\ \end{pmatrix}^{\nu}+\dots\,, \\[2ex] D_{\mu}(G^{\mu i})^{(3)}|_{y^t=0}&=-\frac{n'}{r^2}\widehat{u}^r\widehat{u}^i+\dots\,,\end{aligned}$$ with the normalized three-velocity vector $\widehat{\mathbf{u}}=\mathbf{u}/|\mathbf{u}|$. Comparing the obtained result to reveals that the structure of both expressions is very similar. The difference is a global prefactor of the form $r^2\sqrt{\mathbf{u}^2}$. The dimensionful factor of $r^2$ is not surprising. Both the Riemann curvature tensor and the (modified) Einstein tensor involve two derivatives, which is why their mass dimensions is $-2$. However the energy-momentum tensor is based on the “Lagrangian” of a classical light ray, , which is a dimensionless quantity. The discrepancy in mass dimensions is compensated by the only dimensionful length scale available, which is $r$. It seems that an alternative definition of the Einstein tensor in the framework of Finsler geometry can compensate for the modified energy-momentum conservation law when explicit Lorentz violation is considered. This result is interesting and deserves further study, e.g., whether it holds for anisotropic theories as well. Conclusions and outlook {#sec:conclusion} ======================= In this work classical-ray analogues to the photon sector of the minimal Standard-Model Extension were discussed. It was shown that a nonvanishing photon mass allows for deriving classical point-particle Lagrangians in analogy to the fermion sector. However the standard method used for the fermion sector does not work any more in case the photon mass vanishes. The reason is that a light ray does not have as many degrees of freedom as a massive particle. Instead, for the photon sector an alternative technique had to be employed which allowed to derive a Lagrangian-type function for a classical ray directly from the modified photon dispersion relation. This was carried out for several interesting cases of the minimal, [*CPT*]{}-even photon sector, which is characterized by dimensionless controlling coefficients. Subsequently it was shown that the results obtained are consistent with the eikonal equation approach that describes the geometric-optics limit of an electromagnetic wave. Mathematically the Lagrangian-type functions can be interpreted as Finsler structures. In contrast to the fermion sector they only involve the spatial velocity components and they are closely linked to an effective refractive index of the Lorentz-violating vacuum. It has been known long since that there is a connection between the geodesic equations for a light ray in a gravitational background and the eikonal equations. This link is warranted for weak gravitational fields at least, e.g., in the solar system. It was crucial to set up a phenomenological description of light rays subject to Lorentz violation in a weak gravitational field. This description made it possible to obtain sensitivities on the isotropic controlling coefficient $\widetilde{\kappa}_{\mathrm{tr}}$ that could be probed by the space missions GAIA and LATOR employing measurements of light deflection at massive bodies. The upshot is that the planned mission LATOR may have a sensitivity on $|\widetilde{\kappa}_{\mathrm{tr}}|$ in the order of magnitude of $10^{-16}$ where the running mission GAIA can reach $10^{-14}$. The difference in sensitivity originates from the different precision of measuring angles for both missions. The final part of the paper was dedicated to investigating the properties of the (isotropic) curved spacetime, which the phenomenological studies were based on, from a Finslerian point of view. It was demonstrated that in the classical limit (neglecting higher spacetime derivatives of the refractive index) an Einstein tensor can be defined that is not subject to the usual Bianchi identities in Riemannian geometry. Therefore its covariant derivative is nonzero and it has a form that is related to the modified conservation law of the energy-momentum tensor based on the classical Lagrangian-type function studied in this context. Hence it seems that Finsler geometry provides new geometrical degrees of freedom that can serve as a kind of “buffer” to allow for a momentum transfer whenever the momentum of the light ray changes. These geometrical degrees of freedom take the role of the Nambu-Goldstone modes appearing when spontaneous Lorentz violation is considered. To summarize, the current article provides a technique in treating Lorentz-violating photons in a curved background in a geometric-optics approximation. As an outlook it will be interesting to apply the setup to anisotropic frameworks, first to obtain sensitivities on related controlling coefficients and second to study the properties of the underlying Finsler geometry. Acknowledgments =============== It is a pleasure to thank V. A. Kostelecký for suggesting this line of research and for having fruitful discussions. This work was performed with financial support from the *Deutsche Akademie der Naturforscher Leopoldina* within Grant No. LPDS 2012-17. Classical Lagrangians for massive Lorentz-violating photons {#sec:lagrangians-massive-photons} =========================================================== The first part of the appendix shall briefly demonstrate how to derive the classical Lagrange functions in from the set of equations (\[eq:dispersion-relation-general\]), (\[eq:group-velocity-correspondence\]), and (\[eq:lagrange-function\]). The demonstration will be performed for the nonbirefringent, anisotropic case of the [*CPT*]{}-even sector and for a particular choice of the [*CPT*]{}-odd framework. The calculation is easier for the [*CPT*]{}-even theory, which is why it will be studied first. [*CPT*]{}-even minimal photon sector {#sec:lagrangians-cpt-even-massive-photons} ------------------------------------ The base is where for convenience we set $(3/2)\widetilde{\kappa}_{e-}^{11}\equiv\kappa$. For the remaining [*CPT*]{}-even cases the procedure works analogously. First of all the modified dispersion relation for a massive photon subject to this particular Lorentz-violating framework reads $$(1+\kappa)\left(k_0^2-k_1^2-k_2^2\right)-(1-\kappa)k_3^2=m_{\upgamma}^2\,.$$ To obtain the group velocity components it is often reasonable not to solve the dispersion relation to obtain $k_0$ directly, but to differentiate it implicitly with respect to the spatial momentum components: $$\begin{aligned} 2(1+\kappa)k_0\frac{\partial k_0}{\partial k_1}-2(1+\kappa)k_1=0 &\Leftrightarrow \frac{\partial k_0}{\partial k_1}=\frac{k_1}{k_0}\,, \\[2ex] 2(1+\kappa)k_0\frac{\partial k_0}{\partial k_2}-2(1+\kappa)k_2=0 &\Leftrightarrow \frac{\partial k_0}{\partial k_2}=\frac{k_2}{k_0}\,, \\[2ex] 2(1+\kappa)k_0\frac{\partial k_0}{\partial k_3}-2(1-\kappa)k_3=0 &\Leftrightarrow \frac{\partial k_0}{\partial k_3}=\frac{1-\kappa}{1+\kappa}\frac{k_3}{k_0}\,.\end{aligned}$$ For the particular case studied, leads to the following three equations: $$\frac{k_1}{k_0}=-\frac{u^1}{u^0}\,,\quad \frac{k_2}{k_0}=-\frac{u^2}{u^0}\,,\quad \frac{1-\kappa}{1+\kappa}\frac{k_3}{k_0}=-\frac{u^3}{u^0}\,.$$ Evidently only the third one is modified by Lorentz violation mirroring the spatial anisotropy. These relations can be solved directly to express the spatial momentum components via $k_0$: $$\label{eq:spatial-momentum-components-via-k0} k_1=-\frac{k_0u^1}{u^0}\,,\quad k_2=-\frac{k_0u^2}{u^0}\,,\quad k_3=-\frac{1+\kappa}{1-\kappa}\frac{k_0u^3}{u^0}\,.$$ We can now use and express the spatial momentum components by taking into account the previously obtained results of : $$L=-(k_0u^0+k_1u^1+k_2u^2+k_3u^3)=\frac{k_0}{u^0}\left[-(u^0)^2+(u^1)^2+(u^2)^2+\frac{1+\kappa}{1-\kappa}(u^3)^2\right]\,.$$ The latter is solved with respect to $k_0$ giving an expression comprising the (unknown) Lagrange function and the four-velocity components: $$\label{eq:k0-via-velocity-components} k_0=-L\frac{(1-\kappa)u^0}{(1-\kappa)\left[(u^0)^2-(u^1)^2-(u^2)^2\right]-(1+\kappa)(u^3)^2}\,.$$ Now all four-momentum components in the dispersion relation can be eliminated via and a subsequent insertion of : $$\begin{aligned} 0&=\frac{1+\kappa}{1-\kappa}\frac{k_0^2}{(u^0)^2}\left\{(1-\kappa)\left[(u^0)^2-(u^1)^2-(u^2)^2\right]-(1+\kappa)(u^3)^2\right\}-m_{\upgamma}^2\,, \\[2ex] 0&=L^2\frac{(1-\kappa)(1+\kappa)}{(1-\kappa)\left[(u^0)^2-(u^1)^2-(u^2)^2\right]-(1+\kappa)(u^3)^2}-m_{\upgamma}^2\,.\end{aligned}$$ The final equation comprises a polynomial of the Lagrangian whose coefficients depend on four-velocity components only. The polynomial must be solved to give $L$: $$L^{\pm}=\pm m_{\upgamma}\sqrt{\frac{1}{1+\kappa}\left[(u^0)^2-(u^1)^2-(u^2)^2\right]-\frac{1}{1-\kappa}(u^3)^2}\,.$$ The result corresponds to . The procedure shown is typically applied to derive classical Lagrangians. Four of the five equations are employed to eliminate all four-momentum components and to obtain a polynomial equation in $L$ that only comprises the four-velocity. The latter is then solved with respect to $L$ finally. [*CPT*]{}-odd minimal photon sector {#sec:lagrangians-cpt-odd-massive-photons} ----------------------------------- Due to observer Lorentz invariance without a loss of generality $(k_{AF})^{\kappa}=(0,0,0,1)^{\kappa}$ will be chosen for the spacelike case. The modified dispersion relation involves an isotropic contribution and a second term that does not comprise the momentum component parallel to the preferred spacetime direction: $$\label{eq:dispersion-relation-cpt-odd} (k_0^2-\mathbf{k}^2)^2-4m_{\scriptscriptstyle{\mathrm{CS}}}^2(k_0^2-k_1^2-k_2^2)=0\,.$$ The group velocity components are obtained by implicit differentiation of with respect to the spatial momentum components: $$\begin{aligned} 0&=4(k_0^2-\mathbf{k}^2)\left[k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_1}-k_1\right]-8m_{\scriptscriptstyle{\mathrm{CS}}}^2\left(k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_1}-k_1\right) \notag \\ &=4(k_0^2-\mathbf{k}^2-2m_{\scriptscriptstyle{\mathrm{CS}}}^2)\left[k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_1}-k_1\right]\,, \\[2ex] 0&=4(k_0^2-\mathbf{k}^2-2m_{\scriptscriptstyle{\mathrm{CS}}}^2)\left[k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_2}-k_2\right]\,, \\[2ex] 0&=4(k_0^2-\mathbf{k}^2)\left[k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_3}-k_3\right]-8m_{\scriptscriptstyle{\mathrm{CS}}}^2k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_3} \notag \\ &=4(k_0^2-\mathbf{k}^2-2m_{\scriptscriptstyle{\mathrm{CS}}}^2)k_0\frac{\mathrm{d}k_0}{\mathrm{d}k_3}-4(k_0^2-\mathbf{k}^2)k_3\,.\end{aligned}$$ Since the preferred spacetime direction points along the third axis of the coordinate system, the first and second group velocity components remain standard where only the third one is modified:$$\frac{\mathrm{d}k_0}{\mathrm{d}k_1}=\frac{k_1}{k_0}\,,\quad \frac{\mathrm{d}k_0}{\mathrm{d}k_2}=\frac{k_2}{k_0}\,,\quad \frac{\mathrm{d}k_0}{\mathrm{d}k_3}=\frac{k_3(k_0^2-\mathbf{k}^2)}{k_0(k_0^2-\mathbf{k}^2-2m_{\scriptscriptstyle{\mathrm{CS}}}^2)}\,.$$ Therefore results in $$\label{eq:group-velocity-correspondence-cpt-odd} \frac{k_1}{k_0}=-\frac{u^1}{u^0}\,,\quad \frac{k_2}{k_0}=-\frac{u^2}{u^0}\,,\quad \frac{k_3(k_0^2-\mathbf{k}^2)}{k_0(k_0^2-\mathbf{k}^2-2m_{\scriptscriptstyle{\mathrm{CS}}}^2)}=-\frac{u^3}{u^0}\,.$$ The first two of these relationships allow for writing $k_1$ and $k_2$ in terms of $k_0$. However the third equation would lead to a cumbersome third-order polynomial to be solved, which is not a reasonable step to take. It is better to insert the first two of into and to solve the latter with respect to $k_3$. Then it is possible to express $k_3$ via $k_0$ only: $$k_3=\frac{1}{u^0u^3}\left\{k_0\left[(u^1)^2+(u^2)^2-(u^0)^2\right]-Lu^0\right\}\,.$$ Now we can express all spatial momentum components via $k_0$. Hence we can eliminate all of them in to obtain an equation that only involves $k_0$. This can be solved to write $k_0$ in terms of four-velocity components and the Lagrangian where one of the solutions reads \[eq:k0-solution-cpt-odd\] $$\begin{aligned} k_0&=-u^0\frac{L\sqrt{u_{\bot}^2}+m_{\scriptscriptstyle{\mathrm{CS}}}(u^3)^2+|u^3|\sqrt{L^2+2m_{\scriptscriptstyle{\mathrm{CS}}}\sqrt{u_{\bot}^2}L+m_{\scriptscriptstyle{\mathrm{CS}}}^2(u^3)^2}}{u^2\sqrt{u_{\bot}^2}}\,, \\[2ex] (u_{\bot}^{\mu})&=(u^0,u^1,u^2,0)^T\,.\end{aligned}$$ Here $u^0>0$ has been assumed for simplicity. The last step is to eliminate all four-momentum components in the third of to obtain a polynomial equation for $L$: $$L^2+2m_{\scriptscriptstyle{\mathrm{CS}}}\sqrt{u_{\bot}^2}L+m_{\scriptscriptstyle{\mathrm{CS}}}^2(u^3)^2=0\,,$$ which leads to the Lagrange functions $$L^{\pm}=m_{\scriptscriptstyle{\mathrm{CS}}}\left[\pm\sqrt{u^2}-\sqrt{u_{\bot}^2}\right]\,.$$ Reinstating the preferred spacetime direction, it is possible to write the latter in the form of . Using the other solution of $k_0$ similar to the Lagrangians with the opposite signs are obtained. A computation for $u^0<0$ leads to analogous results. Due to observer Lorentz invariance the form of the Lagrangian stays the same for general spacelike $k_{AF}$. Light deflection in Schwarzschild spacetimes {#sec:light-deflection-schwarzschild} ============================================ In [@Betschart:2008yi] it was found that a constant refractive index $n\neq 1$ due to Lorentz violation leads to a change in light deflection. This result is in contrast to what we obtain from Bouguer’s formula in . A rough explanation is that Bouguer’s formula relies on the eikonal equation, which is equivalent to the null geodesic equations only for a weak gravitational field. However the latter reference is based on a Schwarzschild metric, $$\mathrm{d}\tau^2=\left(1-\frac{2GM}{r}\right)\mathrm{d}t^2-\left(1-\frac{2GM}{r}\right)^{-1}\mathrm{d}r^2-r^2(\mathrm{d}\theta^2+\sin^2\theta\mathrm{d}\phi^2)\,,$$ which in this form is not generally isotropic. To get a more profound understanding, consider the geodesic equations for a photon in a generally isotropic spacetime of with $A=A(r)$. The Christoffel symbols are computed in Riemmanian geometry according to and the geodesic equations read $$\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}+\gamma^{\mu}_{\phantom{\mu}\nu\varrho}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}\lambda}\frac{\mathrm{d}x^{\varrho}}{\mathrm{d}\lambda}=0\,,\quad (x^{\mu})=(t,r,\theta,\phi)^T\,.$$ In what follows, differentiation with respect to the curve parameter $\lambda$ and with respect to $r$, respectively, will be denoted by a dot and a prime. The geodesic equations can then be cast into the following form: \[eq:geodesic-equations-spherically-symmetric-spacetime\] $$\begin{aligned} \label{eq:equation-t} 0&=\ddot{t}-\frac{A'}{A}\dot{r}\dot{t}\,, \displaybreak[0]\\[2ex] \label{eq:equation-r} 0&=\ddot{r}+\frac{A'}{2A}\dot{r}^2-\frac{A'}{2A^3}\dot{t}^2-\left(1+\frac{A'}{2A}r\right)r\dot{\theta}^2-\left(1+\frac{A'}{2A}r\right)r\sin^2(\theta)\dot{\phi}^2\,, \displaybreak[0]\\[2ex] \label{eq:equation-theta} 0&=\ddot{\theta}+\left(\frac{2}{r}+\frac{A'}{A}\right)\dot{r}\dot{\theta}-\sin(\theta)\cos(\theta)\dot{\phi}^2=0\,, \displaybreak[0]\\[2ex] \label{eq:equation-phi} 0&=\ddot{\phi}+\left(\frac{2}{r}+\frac{A'}{A}\right)\dot{r}\dot{\phi}+2\cot(\theta)\dot{\theta}\dot{\phi}=0\,, \displaybreak[0]\\[2ex] \label{eq:equation-null} 0&=\frac{1}{A}\dot{t}^2-A\dot{r}^2-Ar^2\left[\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2\right]\,,\end{aligned}$$ where the fifth of those is the condition for a null-trajectory. They correspond to the equations stated in [@Wu:1988] in case that $A$ is a function of the radial coordinate $r$ only. Now the right-hand side of can be written as the derivative of a conserved quantity that is denoted as $K_0$ in [@Wu:1988]:$$\label{eq:conserved-quantity-1} 0=A\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(\frac{\dot{t}}{A}\right) \Rightarrow K_0=\frac{\dot{t}}{A}\,,\quad \dot{t}=K_0A\,.$$ With the choice of $\theta=\pi/2$ is fulfilled automatically. Using the previous results, can be expressed as the time-derivative of another conserved quantity $K_1$: $$\label{eq:angular-momentum-conservation} 0=\frac{1}{Ar^2}\frac{\mathrm{d}}{\mathrm{d}\lambda}(Ar^2\dot{\phi}) \Rightarrow K_1=Ar^2\dot{\phi}\,,\quad \dot{\phi}=\frac{K_1}{Ar^2}\,.$$ Looking at we see that both $K_0$ and $K_1$ correspond to the conserved quantities that are obtained via the Killing vectors, cf. . From now on the trajectory shall be parameterized with respect to proper time: $\lambda=\tau$. Since $K_0$ is then linked to infinitesimal time translations, it is reasonable to identify it with the total photon energy $E$. Furthermore $K_1$ is connected to infinitesimal changes in the angle $\phi$, which is why it corresponds to the angular momentum $L$. When these conserved quantities are compared to Eqs. (5.7a,b) in [@Betschart:2008yi] we see that the energy is the same, but the angular momentum differs by an additional factor of $A$. Finally can be written as follows: $$\label{eq:conserved-quantity-3} 0=\frac{1}{2A\dot{r}}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(A\dot{r}^2-E^2A+\frac{L^2}{r^2A}\right) \Rightarrow K_2=A\dot{r}^2-E^2A+\frac{L^2}{r^2A}\,.$$ Therefore the latter comprises even another conserved quantity $K_2$. Setting $K_2=0$ is in accordance with the null-trajectory condition of . Taking into account that $\dot{r}=(\mathrm{d}r/\mathrm{d}\phi)\dot{\phi}$ where $\dot{\phi}$ is again expressed by the conserved angular momentum, it is possible to solve with respect to $\mathrm{d}\phi/\mathrm{d}r$: $$\label{eq:change-in-angle} \frac{\mathrm{d}\phi}{\mathrm{d}r}=\frac{L}{r\sqrt{E^2A(r)^2r^2-L^2}}\,,\quad \phi(r)=L\int_{r_0}^{\infty} \frac{\mathrm{d}r}{r\sqrt{E^2A(r)^2r^2-L^2}}\,.$$ Comparing to we see that $C=L/E$, i.e., the constant appearing in Bouguer’s formula can be understood as the ratio of angular momentum and total energy. Now there are some differences between the final result of and the corresponding Eq. (5.9) in [@Betschart:2008yi]. In the latter paper a black-hole gravitational background is considered in Schwarzschild coordinates. This line interval does not have the form of a generally isotropic metric given in . In fact, there are isotropic coordinates allowing us to write the Schwarzschild solution in the form (see, e.g., page 93 of [@Eddington:1923]): $$\begin{aligned} \varrho&=\frac{1}{2}\left(r-GM+\sqrt{r(r-2GM)}\right)\,, \\[2ex] \mathrm{d}\tau^2&=\left(\frac{1-GM/(2\varrho)}{1+GM/(2\varrho)}\right)^2\,\mathrm{d}t^2-\left(1+\frac{GM}{2\varrho}\right)^4\left[\mathrm{d}\varrho^2+\varrho^2\,\mathrm{d}\theta^2+\varrho^2\sin^2\theta\,\mathrm{d}\phi^2\right]\,.\end{aligned}$$ Using this set of coordinates the equation encoding angular momentum conservation and the change in the angle $\phi$ with respect to the new radial coordinate $\rho$ read as follows: $$\begin{aligned} \label{eq:angular-momentum-conservation-isotropic} L&=g_{\rho\rho}\rho^2\dot{\phi}\,, \\[2ex] \frac{\mathrm{d}\phi}{\mathrm{d}\rho}&=\frac{L}{\rho}\frac{\sqrt{g_{\rho\rho}g_{tt}}}{\sqrt{E^2g_{\rho\rho}^2\rho^2-L^2g_{\rho\rho}g_{tt}}}=\frac{L}{\rho}\frac{1}{\sqrt{E^2g_{\rho\rho}^2\rho^2-L^2}}+\mathcal{O}\left(\frac{GM}{2\rho}\right)^2\,. $$ The first corresponds to and the second to neglecting second-order gravity effects. Multiplying the modified line interval of their Eq. (4.12) by the constant $\sqrt{1+\epsilon}$ leads to $$\mathrm{d}\widetilde{\tau}^2=\frac{1}{\sqrt{1+\epsilon}}\left(1-\frac{2GM}{r}\right)\mathrm{d}t^2-\sqrt{1+\epsilon}\left[\frac{1}{1-2GM/r}\mathrm{d}r^2+r^2\,\mathrm{d}\Omega^2\right]\,.$$ Since photons move on null-trajectories, $\mathrm{d}\widetilde{\tau}^2=0$ anyhow, which is why a multiplication of the line element by a constant should not change the physics. In this case the Lorentz-violating contribution governed by a position-independent $\epsilon$ drops out of $\mathrm{d}\phi/\mathrm{d}\rho$ when taking into account . Therefore in the isotropic coordinates the particular Lorentz-violating contribution of their case 3 produces second-order gravity effects associated to Lorentz violation. Far away from the back-hole event horizon there are no novel physical effects and this corresponds to the outcome of the eikonal approach. Killing vectors of a spherically symmetric spacetime {#sec:killing-vectors-generally-isotropic} ---------------------------------------------------- In the current paragraph the Killing vectors for a spherically symmetric spacetime, cf.  with $A(r,\theta,\phi)=A(r)$, will be listed. The Killing vectors $\xi_{\mu}$ describe infinitesimal isometries for a spacetime and they are linked to underlying symmetries and conserved quantities. In general they are obtained from a set of partial differential equations called the Killing equations: $$D_{\alpha}\xi_{\beta}+D_{\beta}\xi_{\alpha}=0\,,\quad D_{\nu}\xi_{\lambda}=\partial_{\nu}\xi_{\lambda}-\gamma^{\mu}_{\phantom{\mu}\nu\lambda}\xi_{\mu}\,,$$ with the covariant derivative $D_{\alpha}$ and the Christoffel symbols $\gamma^{\mu}_{\phantom{\mu}\nu\lambda}$. The latter can be directly extracted from . For the spherically symmetric spacetime it is possible to solve the Killing equations analytically. To do so, it is reasonable to make a certain *Ansatz*, e.g., one with vanishing spatial components of $\xi_{\mu}$. This simplifies the set of equations dramatically where several are immediately fulfilled automatically. They are then solved successively to obtain four Killing vectors. Since the metric is isotropic, it is reasonable to make an *Ansatz* for $\xi_{\mu}$ that only involves a nonvanishing timelike component that does not depend on time itself: $$(\xi_{\mu})=\begin{pmatrix} \xi_0(r,\theta,\phi) \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}\,.$$ In this case the following three differential equations must to be solved: $$\xi_0\frac{A'}{A}+\frac{\partial\xi_0}{\partial r}=0\,,\quad \frac{\partial\xi_0}{\partial\theta}=0\,,\quad \frac{\partial\xi_0}{\partial\phi}=0\,.$$ The remaining ones are fulfilled automatically. The latter two tell us immediately that $\xi_0$ neither depends on $\theta$ nor $\phi$. Therefore the first differential equation is an ordinary one that can be solved directly by integration: $$\frac{\xi_0'}{\xi_0}=-\frac{A'}{A} \Rightarrow \ln|\xi_0|=-\ln|c_0A| \Rightarrow \xi_0(r)=\frac{\widetilde{c}_0}{A(r)}\,,\quad c_0,\,\widetilde{c}_0\in \mathbb{R}\,.$$ A similar approach leads to the remaining Killing vectors. In total one obtains $$\begin{aligned} \xi_{\mu}^{(1)}&=\begin{pmatrix} 1/A \\ 0 \\ 0 \\ 0 \\ \end{pmatrix}\,,\quad \xi_{\mu}^{(2)}=r^2A\begin{pmatrix} 0 \\ 0 \\ \sin\phi \\ \sin\theta\cos\theta\cos\phi \\ \end{pmatrix}\,, \displaybreak[0]\\[2ex] \xi_{\mu}^{(3)}&=r^2A\begin{pmatrix} 0 \\ 0 \\ \cos\phi \\ -\sin\theta\cos\theta\sin\phi \\ \end{pmatrix}\,,\quad \xi_{\mu}^{(4)}=r^2A\begin{pmatrix} 0 \\ 0 \\ 0 \\ \sin^2\theta \\ \end{pmatrix}\,.\end{aligned}$$ Suitable contractions of the Killing vectors with $(\dot{x}^{\mu})=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})^T$ (and additional linear combinations) lead to conserved quantities. The first conserved quantity follows from a contraction with the first Killing vector: \[eq:conserved-quantities-killing\] $$\label{eq:conserved-quantities-killing-1} \xi_{\mu}^{(1)}\dot{x}^{\mu}=\frac{\dot{t}}{A}=\mathrm{const.}$$ The latter corresponds to the result obtained in and it is related to energy conservation. The second conserved quantity involves the remaining Killing vectors where it is understood to be evaluated at $\theta=\pi/2$: $$\label{eq:conserved-quantities-killing-2} \sqrt{\sum_{i=2}^4 \left.(\xi_{\mu}^{(i)}\dot{x}^{\mu})^2\right|_{\theta=\pi/2}}=Ar^2\dot{\phi}=\mathrm{const.}$$ This conserved quantity is the same as what was obtained in and it means angular momentum conservation. Hence the Killing vectors $\xi^{(i)}$ for $i=2\dots 4$ are related to rotational symmetry of the spherically symmetric spacetime. Eikonal equation for inhomogeneous and anisotropic media {#sec:eikonal-equation-inhomogeneous-anisotropic} ======================================================== The current section serves with providing some general results on the physics of the eikonal equation, which are used in extensively. In general, the eikonal equation provides a set of three coupled nonlinear differential equations. In what follows a refractive index bearing a dependence on the radial distance $r$ and an angle $\phi$ is assumed (cf., e.g., ). The photon trajectory shall be parameterized by the angle $\phi$, i.e., $\mathbf{r}(\phi)=r(\phi)\widehat{\mathbf{e}}_r(\phi)$. Its first and second derivative read $$\label{eq:photon-trajectory-derivatives} \mathbf{r}'=\dot{r}\widehat{\mathbf{e}}_r+r\widehat{\mathbf{e}}_{\phi}\,,\quad \mathbf{r}''=(\ddot{r}-r)\widehat{\mathbf{e}}_r+2\dot{r}\widehat{\mathbf{e}}_{\phi}\,.$$ The arc length depends on $\phi$ and we obtain a set of useful relationships: $$\begin{aligned} \label{eq:eikonal-equation-identities} s(\phi)&=\int^{\phi} \mathrm{d}\phi'\,|\mathbf{r}'|=\int^{\phi} \mathrm{d}\phi'\,\sqrt{r^2+\dot{r}^2}\,,\quad \frac{\mathrm{d}s}{\mathrm{d}\phi}=\sqrt{r^2+\dot{r}^2}\,, \\[2ex] \left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-2}&=\frac{1}{r^2+\dot{r}^2}\,,\quad \frac{\mathrm{d}}{\mathrm{d}\phi}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}=\frac{\mathrm{d}}{\mathrm{d}\phi}\left(\frac{1}{\sqrt{r^2+\dot{r}^2}}\right)=-\frac{(r+\ddot{r})\dot{r}}{(r^2+\dot{r}^2)^{3/2}}\,, \\[2ex] \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}&=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}\,.\end{aligned}$$ Now the derivative on the left-hand side of the eikonal equation can be computed. Instead of differentiating with respect to the arc length we have to calculate derivatives with respect to $\phi$, which leads to three terms: $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s}\left[n\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}\right]&=\frac{\mathrm{d}n}{\mathrm{d}s}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}+n\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}\phi^2}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-2}+\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}n\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi} \notag \displaybreak[0]\\ &=\frac{\mathrm{d}n}{\mathrm{d}\phi}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-2}+n\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}\phi^2}\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-2} \notag \displaybreak[0]\\ &\phantom{{}={}}+\frac{\mathrm{d}}{\mathrm{d}\phi}\left[\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}\right]\left(\frac{\mathrm{d}s}{\mathrm{d}\phi}\right)^{-1}n\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\phi}\,.\end{aligned}$$ Now employing the derivatives of and the identities given in the eikonal equation can be expressed in terms of the radial coordinate $r$, the angle $\phi$, and the basis vectors: $$\begin{aligned} \widehat{\mathbf{e}}_r\frac{\partial n}{\partial r}+\frac{1}{r}\frac{\partial n}{\partial\phi}\widehat{\mathbf{e}}_{\phi}&=\frac{1}{r^2+\dot{r}^2}\left\{\frac{\mathrm{d}n}{\mathrm{d}\phi}(\dot{r}\widehat{\mathbf{e}}_r+r\widehat{\mathbf{e}}_{\phi})+n\left[(\ddot{r}-r)\widehat{\mathbf{e}}_r+2\dot{r}\widehat{\mathbf{e}}_{\phi}\right]\right\} \notag \\ &\phantom{{}={}}-n(\dot{r}\widehat{\mathbf{e}}_r+r\widehat{\mathbf{e}}_{\phi})\frac{(r+\ddot{r})\dot{r}}{(r^2+\dot{r}^2)^2}\end{aligned}$$ Sorting terms associated to $\widehat{\mathbf{e}}_r$ and $\widehat{\mathbf{e}}_{\phi}$, respectively, results in a system of two differential equations: $$\begin{aligned} \frac{\partial n}{\partial r}&=\frac{1}{r^2+\dot{r}^2}\left[\frac{\mathrm{d}n}{\mathrm{d}\phi}\dot{r}+n(\ddot{r}-r)\right]-n\frac{(r+\ddot{r})\dot{r}^2}{(r^2+\dot{r}^2)^2}\,, \\[2ex] \frac{1}{r}\frac{\partial n}{\partial\phi}&=\frac{1}{r^2+\dot{r}^2}\left[\frac{\mathrm{d}n}{\mathrm{d}\phi}r+2n\dot{r}\right]-n\frac{(r+\ddot{r})r\dot{r}}{(r^2+\dot{r}^2)^2}\,.\end{aligned}$$ Multiplying the second with $\dot{r}/r$ and subtracting it from the first eliminates various terms, which simplifies the equation drastically: $$\begin{aligned} \frac{\partial n}{\partial r}-\frac{\dot{r}}{r^2}\frac{\partial n}{\partial\phi}&=\frac{n}{r^2+\dot{r}^2}\left(\ddot{r}-\frac{2\dot{r}^2}{r}-r\right)\,, \\[2ex] \label{eq:eikonal-equation-main-result} 0&=(r^2+\dot{r}^2)\left[r\frac{\partial n}{\partial r}-\frac{\dot{r}}{r}\frac{\partial n}{\partial\phi}\right]+n(r^2+2\dot{r}^2-r\ddot{r})\,.\end{aligned}$$ This is the final result that we are interested in and that shall be used for practical purposes. However multiplying the latter with $r\dot{r}/(r^2+\dot{r}^2)^{3/2}$, it can be written in a form that allows for a deeper physical understanding: $$\frac{\mathrm{d}}{\mathrm{d}\phi}(nr\sin\alpha)-\sqrt{r^2+\dot{r}^2}\frac{\partial n}{\partial\phi}=0\,,\quad \sin\alpha=\frac{r}{\sqrt{r^2+\dot{r}^2}}\,.$$ If the refractive index only depends on the radial coordinate the second term on the left-hand side vanishes, which then leads us directly to the formula of Bouguer. Physically this result means angular momentum conservation. For a refractive index that additionally depends on the angle $\phi$ angular momentum is not a conserved quantity any more. Instead, there is a driving term that modifies the angular momentum. The change is bigger the larger the velocity is and the stronger the refractive index changes with the angle. [99]{} D. Colladay and V. A. Kostelecký, “Lorentz violating extension of the standard model,” Phys. Rev. D [**58**]{}, 116002 (1998), hep-ph/9809521. V. A. Kostelecký and M. Mewes, “Electrodynamics with Lorentz-violating operators of arbitrary dimension,” Phys. Rev. D [**80**]{}, 015020 (2009), arXiv:0905.0031 \[hep-ph\]. V. A. Kostelecký and M. Mewes, “Neutrinos with Lorentz-violating operators of arbitrary dimension,” Phys. Rev. D [**85**]{}, 096005 (2012), arXiv:1112.6395 \[hep-ph\]. V. A. Kostelecký and M. Mewes, “Fermions with Lorentz-violating operators of arbitrary dimension,” Phys. Rev. D [**88**]{}, 096006 (2013), arXiv:1308.4973 \[hep-ph\]. V. A. Kostelecký and N. Russell, “Data Tables for Lorentz and *CPT* Violation,” Rev. Mod. Phys.  [**83**]{}, 11 (2011), arXiv:0801.0287 \[hep-ph\]. O. W. Greenberg, “CPT violation implies violation of Lorentz invariance,” Phys. Rev. Lett. [**89**]{}, 231602 (2002), hep-ph/0201258. V. A. Kostelecký and S. Samuel, “Spontaneous breaking of Lorentz symmetry in string theory,” Phys. Rev. D [**39**]{}, 683 (1989). V. A. Kostelecký and R. Potting, “*CPT* and strings,” Nucl. Phys. B [**359**]{}, 545 (1991). V. A. Kostelecký and R. Potting, “*CPT*, strings, and meson factories,” Phys. Rev. D [**51**]{}, 3923 (1995), hep-ph/9501341. R. Gambini and J. Pullin, “Nonstandard optics from quantum space-time,” Phys. Rev. D [**59**]{}, 124021 (1999), gr-qc/9809038. M. Bojowald, H. A. Morales–Técotl, and H. Sahlmann, “Loop quantum gravity phenomenology and the issue of Lorentz invariance,” Phys. Rev. D [**71**]{}, 084012 (2005), gr-qc/0411101. G. Amelino-Camelia and S. Majid, “Waves on noncommutative spacetime and gamma-ray bursts,” Int. J. Mod. Phys. A [**15**]{}, 4301 (2000), hep-th/9907110. S. M. Carroll, J. A. Harvey, V. A. Kostelecký, C. D. Lane, and T. Okamoto, “Noncommutative field theory and Lorentz violation,” Phys. Rev. Lett. [**87**]{}, 141601 (2001), hep-th/0105082. F. R. Klinkhamer and C. Rupp, “Spacetime foam, *CPT* anomaly, and photon propagation,” Phys. Rev. D [**70**]{}, 045020 (2004), hep-th/0312032. S. Bernadotte and F. R. Klinkhamer, “Bounds on length scales of classical spacetime foam models,” Phys. Rev. D [**75**]{}, 024028 (2007), hep-ph/0610216. S. Hossenfelder, “Theory and phenomenology of space-time defects,” Adv. High Energy Phys. [**2014**]{}, 950672 (2014), arXiv:1401.0276 \[hep-ph\]. F. R. Klinkhamer, “Z-string global gauge anomaly and Lorentz non-invariance,” Nucl. Phys. B [**535**]{}, 233 (1998), hep-th/9805095. F. R. Klinkhamer, “A *CPT* anomaly,” Nucl. Phys. B [**578**]{}, 277 (2000), hep-th/9912169. V. A. Kostelecký and R. Lehnert, “Stability, causality, and Lorentz and [*CPT*]{} violation,” Phys. Rev. D [**63**]{}, 065008 (2001), hep-th/0012060. C. Adam and F. R. Klinkhamer, “Causality and *CPT* violation from an Abelian Chern-Simons-like term,” Nucl. Phys. B [**607**]{}, 247 (2001), hep-ph/0101087. R. Casana, M. M. Ferreira, A. R. Gomes, and P. R. D. Pinheiro, “Gauge propagator and physical consistency of the *CPT*-even part of the standard model extension,” Phys. Rev. D [**80**]{}, 125040 (2009), arXiv:0909.0544 \[hep-th\]. R. Casana, M. M. Ferreira, A. R. Gomes, and F. E. P. dos Santos, “Feynman propagator for the nonbirefringent *CPT*-even electrodynamics of the standard model extension,” Phys. Rev. D [**82**]{}, 125006 (2010), arXiv:1010.2776 \[hep-th\]. F. R. Klinkhamer and M. Schreck, “Consistency of isotropic modified Maxwell theory: Microcausality and unitarity,” Nucl. Phys. B [**848**]{}, 90 (2011), arXiv:1011.4258 \[hep-th\], F. R. Klinkhamer and M. Schreck, “Models for low-energy Lorentz violation in the photon sector: Addendum to ‘Consistency of isotropic modified Maxwell theory’,” Nucl. Phys. B [**856**]{}, 666 (2012), arXiv:1110.4101 \[hep-th\]. M. Schreck, “Analysis of the consistency of parity-odd nonbirefringent modified Maxwell theory,” Phys. Rev. D [**86**]{}, 065038 (2012), arXiv:1111.4182 \[hep-th\]. M. Cambiaso, R. Lehnert, and R. Potting, “Massive photons and Lorentz violation,” Phys. Rev. D [**85**]{}, 085023 (2012), arXiv:1201.3045 \[hep-th\]. D. Colladay, P. McDonald, and R. Potting, “Gupta-Bleuler photon quantization in the standard model extension,” Phys. Rev. D [**89**]{}, 085014 (2014), arXiv:1401.1173 \[hep-ph\]. M. Maniatis and C. M. Reyes, “Unitarity in a Lorentz symmetry breaking model with higher-order operators,” Phys. Rev. D [**89**]{}, 056009 (2014), arXiv:1401.3752 \[hep-ph\]. M. Schreck, “Quantum field theory based on birefringent modified Maxwell theory,” Phys. Rev. D [**89**]{}, 085013 (2014), arXiv:1311.0032 \[hep-th\]. M. Schreck, “Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the photon sector,” Phys. Rev. D [**89**]{}, 105019 (2014), arXiv:1312.4916 \[hep-th\]. M. Cambiaso, R. Lehnert, and R. Potting, “Asymptotic states and renormalization in Lorentz-violating quantum field theory,” Phys. Rev. D [**90**]{}, 065003 (2014), arXiv:1401.7317 \[hep-th\]. M. Schreck, “Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the fermion sector,” Phys. Rev. D [**90**]{}, 085025 (2014), arXiv:1403.6766 \[hep-th\]. S. Albayrak and I. Turan, “[*CPT*]{}-odd photon in vacuum-orthogonal model,” arXiv:1505.07584 \[hep-ph\]. V. A. Kostelecký, C. D. Lane, and A. G. M. Pickering, “One-loop renormalization of Lorentz-violating electrodynamics,” Phys. Rev. D [**65**]{}, 056006 (2002), hep-th/0111123. D. Colladay and P. McDonald, “One-loop renormalization of pure Yang-Mills theory with Lorentz violation,” Phys. Rev. D [**75**]{}, 105002 (2007), hep-ph/0609084. D. Colladay and P. McDonald, “One-Loop renormalization of QCD with Lorentz violation,” Phys. Rev. D [**77**]{}, 085006 (2008), arXiv:0712.2055 \[hep-ph\]. D. Colladay and P. McDonald, “One-Loop renormalization of the electroweak sector with Lorentz violation,” Phys. Rev. D [**79**]{}, 125019 (2009), arXiv:0904.1219 \[hep-ph\]. T. R. S. Santos and R. F. Sobreiro, “Remarks on the renormalization properties of Lorentz and CPT violating quantum electrodynamics,” arXiv:1502.06881 \[hep-th\]. T. R. S. Santos and R. F. Sobreiro, “Renormalizability of Yang-Mills theory with Lorentz violation and gluon mass generation,” Phys. Rev. D [**91**]{}, 025008 (2015), arXiv:1404.4846 \[hep-th\]. V. A. Kostelecký, “Gravity, Lorentz violation, and the standard model,” Phys. Rev. D [**69**]{}, 105009 (2004), hep-th/0312310. Q. G. Bailey and V. A. Kostelecký, “Signals for Lorentz violation in post-Newtonian gravity,” Phys. Rev. D [**74**]{}, 045001 (2006), gr-qc/0603030. V. A. Kostelecký, N. Russell, and J. D. Tasson, “Constraints on torsion from bounds on Lorentz violation,” Phys. Rev. Lett. [**100**]{}, 111102 (2008), arXiv:0712.4393 \[gr-qc\]. V. A. Kostelecký and J. D. Tasson, “Prospects for large relativity violations in matter-gravity couplings,” Phys. Rev. Lett. [**102**]{}, 010402 (2009), arXiv:0810.1459 \[gr-qc\]. Q. G. Bailey, “Time-delay and Doppler tests of the Lorentz symmetry of gravity,” Phys. Rev. D [**80**]{}, 044004 (2009), arXiv:0904.0278 \[gr-qc\]. V. A. Kostelecký and J. D. Tasson, “Matter-gravity couplings and Lorentz violation,” Phys. Rev. D [**83**]{}, 016013 (2011), arXiv:1006.4106 \[gr-qc\]. J. D. Tasson, “Gravitational physics with antimatter,” Hyperfine Interact. [**193**]{}, 291 (2009), arXiv:1010.2811 \[hep-ph\]. J. D. Tasson, “Lorentz violation, gravitomagnetism, and intrinsic spin,” Phys. Rev. D [**86**]{}, 124021 (2012), arXiv:1211.4850 \[hep-ph\]. J. D. Tasson, “Antimatter, the SME, and gravity,” Hyperfine Interact. [**213**]{}, 137 (2012), arXiv:1212.1636 \[hep-ph\]. Y. Bonder, “Lorentz violation in a uniform Newtonian gravitational field,” Phys. Rev. D [**88**]{}, 105011 (2013), arXiv:1309.7972 \[hep-ph\]. Y. Bonder, “Lorentz violation in the gravity sector: The $t$ puzzle,” Phys. Rev. D [**91**]{}, 125002 (2015), arXiv:1504.03636 \[gr-qc\]. Q. G. Bailey, V. A. Kostelecký, and R. Xu, “Short-range gravity and Lorentz violation,” Phys. Rev. D [**91**]{}, 022006 (2015), arXiv:1410.6162 \[gr-qc\]. J. C. Long and V. A. Kostelecký, “Search for Lorentz violation in short-range gravity,” Phys. Rev. D [**91**]{}, 092003 (2015), arXiv:1412.8362 \[hep-ex\]. V. A. Kostelecký and S. Samuel, “Phenomenological gravitational constraints on strings and higher-dimensional theories,” Phys. Rev. Lett. [**63**]{}, 224 (1989). V. A. Kostelecký and S. Samuel, “Gravitational phenomenology in higher-dimensional theories and strings,” Phys. Rev. D [**40**]{}, 1886 (1989). R. Bluhm, N. L. Gagne, R. Potting, and A. Vrublevskis, “Constraints and stability in vector theories with spontaneous Lorentz violation,” Phys. Rev. D [**77**]{}, 125007 (2008) \[Erratum-ibid. D [**79**]{}, 029902 (2009)\], arXiv:0802.4071 \[hep-th\]. C. Hernaski, “Quantization and stability of bumblebee electrodynamics,” Phys. Rev. D [**90**]{}, 124036 (2014), arXiv:1411.5321 \[hep-th\]. R. Bluhm, “Explicit versus spontaneous diffeomorphism breaking in gravity,” Phys. Rev. D [**91**]{}, 065034 (2015), arXiv:1401.4515 \[gr-qc\]. P. Finsler, [*Über Kurven und Flächen in allgemeinen Räumen*]{}, in German (Gebr. Leemann & Co., Zürich, 1918). É. Cartan, Sur les espaces de Finsler, in French, C. R. Acad. Sci. (Paris) [**196**]{}, 582 (1933). D. Bao, S.-S. Chern, and Z. Shen, [*An Introduction to Riemann-Finsler Geometry*]{} (Springer, New York, 2000). P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, [*The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology*]{} (Springer Science + Business Media, Dordrecht, 1993). V. A. Kostelecký and N. Russell, “Classical kinematics for Lorentz violation,” Phys. Lett. B [**693**]{}, 443 (2010), arXiv:1008.5062 \[hep-ph\]. V. A. Kostelecký, “Riemann-Finsler geometry and Lorentz-violating kinematics,” Phys. Lett. B [**701**]{}, 137 (2011), arXiv:1104.5488 \[hep-th\]. V. A. Kostelecký, N. Russell, and R. Tso, “Bipartite Riemann-Finsler geometry and Lorentz violation,” Phys. Lett. B [**716**]{}, 470 (2012), arXiv:1209.0750 \[hep-th\]. D. Colladay and P. McDonald, “Classical Lagrangians for momentum dependent Lorentz violation,” Phys. Rev. D [**85**]{}, 044042 (2012), arXiv:1201.3931 \[hep-ph\]. N. Russell, “Finsler-like structures from Lorentz-breaking classical particles,” Phys. Rev. D [**91**]{}, 045008 (2015), arXiv:1501.02490 \[hep-th\]. M. Schreck, “Classical kinematics and Finsler structures for nonminimal Lorentz-violating fermions,” Eur. Phys. J. C [**75**]{}, 187 (2015), arXiv:1405.5518 \[hep-th\]. M. Schreck, “Classical kinematics for isotropic, minimal Lorentz-violating fermion operators,” Phys. Rev. D [**91**]{}, 105001 (2015), arXiv:1409.1539 \[hep-th\]. J. E. G. Silva and C. A. S. Almeida, “Kinematics and dynamics in a bipartite-Finsler spacetime,” Phys. Lett. B [**731**]{}, 74 (2014), arXiv:1312.7369 \[hep-th\]. J. Foster and R. Lehnert, “Classical-physics applications for Finsler $b$ space,” Phys. Lett. B [**746**]{}, 164 (2015), arXiv:1504.07935 \[physics.class-ph\]. D. Colladay and P. McDonald, “Singular Lorentz-violating Lagrangians and associated Finsler structures,” arXiv:1507.01018 \[hep-ph\]. H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero: I,” Ann. of Math. [**79**]{}, 109, “Resolution of singularities of an algebraic variety over a field of characteristic zero: II,” Ann. of Math. [**79**]{}, 205 (1964). V. A. Kostelecký and M. Mewes, “Signals for Lorentz violation in electrodynamics,” Phys. Rev. D [**66**]{}, 056005 (2002), hep-ph/0205211. Q. G. Bailey and V. A. Kostelecký, “Lorentz-violating electrostatics and magnetostatics,” Phys. Rev. D [**70**]{}, 076006 (2004), arXiv:hep-ph/0407252. S. M. Carroll, G. B. Field, and R. Jackiw, “Limits on a Lorentz- and parity-violating modification of electrodynamics,” Phys. Rev. D [**41**]{}, 1231 (1990). Z. Shen, “Landsberg curvature, $S$-curvature and Riemann curvature,” in [*A Sampler of Riemann-Finsler Geometry*]{}, edited by D. Bao, R. L. Bryant, S.-S. Chern, and Z. Shen, Math. Sci. Res. Inst. Publ. [**50**]{} (Cambridge University Press, Cambridge $\cdot$ New York $\cdot$ Port Melbourne $\cdot$ Madrid $\cdot$ Cape Town, 2004). A. Deicke, “Über die Finsler-Räume mit $A_i=0$,” in German, Arch. Math. [**4**]{}, 45 (1953). M. Matsumoto and S. Hōjō, “A conclusive theorem on C-reducible Finsler spaces,” Tensor (N.S.) [**32**]{}, 225 (1978). N. McGinnis, “Finsler structures in the photon sector of the Standard Model Extension,” Bachelor thesis, unpublished (2014). B. Altschul, “Vacuum Čerenkov radiation in Lorentz-violating theories without *CPT* violation,” Phys. Rev. Lett. [**98**]{}, 041603 (2007), hep-th/0609030. V. Fock, [*The Theory of Space Time and Gravitation*]{}, translated from Russian by N. Kemmer (Pergamon Press, New York $\cdot$ London $\cdot$ Paris $\cdot$ Los Angeles, 1959). O. Constantinescu and M. Crasmareanu, Examples of conics arising in two-dimensional Finsler and Lagrange geometries, An. St. Uni. Mat. [**17**]{}, 45 (2009). G. Randers, “On an asymmetrical metric in the four-space of General Relativity,” Phys. Rev. [**59**]{}, 195 (1941). V. Perlick, “Fermat principle in Finsler spacetimes,” Gen. Rel. Grav.  [**38**]{}, 365 (2006), gr-qc/0508029. R. G. Torromé, P. Piccione, and H. Vitório, “On Fermat’s principle for causal curves in time oriented Finsler spacetimes,” J. Math. Phys. [**53**]{}, 123511 (2012), arXiv:1202.3869 \[math.DG\]. Z. Xiao, L. Shao, and B. Q. Ma, “Eikonal equation of the Lorentz-violating Maxwell theory,” Eur. Phys. J. C [**70**]{}, 1153 (2010) arXiv:1011.5074 \[hep-th\]. X.-J. Wu and C.-M. Xu, “Null geodesic equation equivalent to the geometric optics equation,” Commun. in Theor. Phys. [**9**]{}, 119 (1988). G. Betschart, E. Kant, and F. R. Klinkhamer, “Lorentz violation and black-hole thermodynamics,” Nucl. Phys. B [**815**]{}, 198 (2009), arXiv:0811.0943 \[hep-th\]. M. Born and E. Wolf, *Electromagnetic theory of propagation interference and diffraction of light*, 7th ed. (Cambridge University Press, Cambridge, England, 2005). I. S. Gradshteyn and I. M. Ryzhik, *Table of Integrals, Series, and Products*, 7nd ed. (Academic Press, Burlington $\cdot$ San Diego $\cdot$ London, 2007). D. Lynden-Bell and J. Katz, “Gravitational field energy density for spheres and black holes,” Mon. Not. R. astr. Soc. [**213**]{}, 21 (1985). [@Klinkhamer:2003ec] F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of May 29, 1919,” Phil. Trans. Roy. Soc. A [**220**]{}, 291 (1920). A. S. Eddington, [*The Mathematical Theory of Relativity*]{} (Cambridge University Press, London, 1923). M. Abramowitz and I. A. Stegun, [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*]{} (National Bureau of Standards, Washington, DC, 1964). M. A. C. Perryman, K. S. de Boer, G. Gilmore, E. Høg, M. G. Lattanzi, L. Lindegren, X. Luri, F. Mignard, O. Pace, and P. T. de Zeeuw, “GAIA: Composition, formation and evolution of the galaxy,” Astron. Astrophys. [**369**]{}, 339 (2001), astro-ph/0101235. S. G. Turyshev, M. Shao, and K. L. Nordtvedt, Jr., “Experimental design for the LATOR mission,” Int. J. Mod. Phys. D [**13**]{}, 2035 (2004), gr-qc/0410044. S. G. Turyshev, M. Shao, K. L. Nordtvedt, Jr., H. Dittus, C. Lämmerzahl, S. Theil, C. Salomon, and S. Reynaud [*et al.*]{}, “Advancing fundamental physics with the Laser Astrometric Test of Relativity,” Exper. Astron. [**27**]{}, 27 (2009). J. Evans and M. Rosenquist, “‘$F=ma$’ optics,” Am. J. Phys. [**54**]{}, 876 (1986). F. J. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica [**7**]{}, 449 (1940). H. Watanabe and H. Murayama, “Redundancies in Nambu-Goldstone bosons,” Phys. Rev. Lett. [**110**]{}, 181601 (2013), arXiv:1302.4800 \[cond-mat.other\]. [^1]: Note that the global prefactor of MCS-theory is different in the latter reference. [^2]: The acronym originally meant “Global Astrometric Interferometer for Astrophysics.” Although the foreseen measurement technique was changed upon construction of the apparatus, the acronym was kept. [^3]: The number of spontaneously broken translational and rotational generators is six. However as they are not independent from each other, the total number of Nambu-Goldstone bosons is reduced by these constraints to be three. Relations between broken symmetries and Nambu-Goldstone bosons in certain nonrelativistic systems such as crystals, ferromagnets, and superfluids are nicely described in [@Watanabe:2013iia].

Q: Reset backgroundTimeRemaining when app goes to background Using the following code I am able to run my background app for around 180 seconds. My final objective is to keep it running forever. Battery/processor overuse is not a concern for me as I'm making this for myself. This is code I'm using to run my app in the background. UIApplication * application = [UIApplication sharedApplication]; if([[UIDevice currentDevice] respondsToSelector:@selector(isMultitaskingSupported)]) { NSLog(@"Multitasking Supported"); __block UIBackgroundTaskIdentifier background_task; background_task = [application beginBackgroundTaskWithExpirationHandler:^ { //Clean up code. Tell the system that we are done. [application endBackgroundTask: background_task]; background_task = UIBackgroundTaskInvalid; }]; //To make the code block asynchronous dispatch_async(dispatch_get_global_queue(DISPATCH_QUEUE_PRIORITY_DEFAULT, 0), ^{ //### background task starts NSLog(@"Running in the background\n"); while(TRUE) { NSLog(@"Background time Remaining: %f",[[UIApplication sharedApplication] backgroundTimeRemaining]); [NSThread sleepForTimeInterval:1]; //wait for 1 sec } //#### background task ends //Clean up code. Tell the system that we are done. [application endBackgroundTask: background_task]; background_task = UIBackgroundTaskInvalid; }); } else { NSLog(@"Multitasking Not Supported"); } Background time remaining: starts at around 180 seconds and by the time it comes down to 5, the app gets suspended. No matter what I try, this cannot be avoided. My device is running iOS 8.1 and here are the methods I've tried so far: Method 1: I set background modes as location in info.plist and used the following code as shown in this tutorial. CLLocationManager * manager = [CLLocationManager new]; __block UIBackgroundTaskIdentifier background_task; and then [manager startUpdatingLocation]; Method 2: Using the following lines of code before the app enters the background state. CLLocationManager* locationManager = [[CLLocationManager alloc] init]; [locationManager startUpdatingLocation]; I just need to reset backgroundTimeRemaining and any hack will be fine. Thank you all. A: The way I solved this problem was following the updated part of this tutorial.

Siawosch Azadi In Persia rural carpets have been made in nearly every possible technical variation and for a wide range of uses. Yet there are many nomadic groups whose works are absolutely unknown, and the weavings of other groups have been only very imperfectly studied and described. Walter Denny These include those woven in the former Turkmen, Uzbek, Tajik, Karakalpak Autonomous, Kirgiz, and Kazakh Soviet Socialist Republics; extreme northern and northeastern Persia; Afghanistan; and the Turkic (Uighur) areas of Sinkiang (Xinjiang) in western China. Cross-Reference Richard W. Cottam (1977-81): POLICY TOWARD PERSIA. When the administration of President Jimmy Carter took office in January 1977, United States foreign relations overall were remarkably stable. A modus vivendi had been established with the Soviet Union. Fridrik Thordarson Ehsan Yarshater village in the mountainous area of the Upper Ṭārom district (baḵš) in the šahrestān of Zanjān, at 49°1′ E, 36°52′ N, 42 km north of the district center, Sīrdān. It is one of the few villages in Ṭārom where Iranian Tati dialects have not yet given way to Turkish.

Alpha-ketoglutarate modulates the circadian patterns of lipid peroxidation and antioxidant status during N-nitrosodiethylamine-induced hepatocarcinogenesis in rats. The effect of alpha-ketoglutarate (alpha-KG) on circadian patterns of lipid peroxides and antioxidants in N-nitrosodiethylamine (NDEA)-induced hepatocarcinogenesis in rats has been studied. The circadian rhythm characteristics (acrophase, amplitude, and mesor) of thiobarbituric acid-reactive substances (TBARS), antioxidants, superoxide dismutase, catalase, glutathione peroxidase, and reduced glutathione were markedly altered in NDEA-treated rats. The delays in acrophase observed in NDEA-treated rats were brought back to near normal range by the administration of alpha-KG. An increase in mesor values of TBARS and a decrease in mesor values of antioxidants in NDEA-administered rats were reversed by alpha-KG administration. It can be concluded that alpha-KG exerts its chemopreventive effect by restoring antioxidants and their circadian rhythms.

package match import ( "fmt" "strings" ) type PrefixSuffix struct { Prefix, Suffix string } func NewPrefixSuffix(p, s string) PrefixSuffix { return PrefixSuffix{p, s} } func (self PrefixSuffix) Index(s string) (int, []int) { prefixIdx := strings.Index(s, self.Prefix) if prefixIdx == -1 { return -1, nil } suffixLen := len(self.Suffix) if suffixLen <= 0 { return prefixIdx, []int{len(s) - prefixIdx} } if (len(s) - prefixIdx) <= 0 { return -1, nil } segments := acquireSegments(len(s) - prefixIdx) for sub := s[prefixIdx:]; ; { suffixIdx := strings.LastIndex(sub, self.Suffix) if suffixIdx == -1 { break } segments = append(segments, suffixIdx+suffixLen) sub = sub[:suffixIdx] } if len(segments) == 0 { releaseSegments(segments) return -1, nil } reverseSegments(segments) return prefixIdx, segments } func (self PrefixSuffix) Len() int { return lenNo } func (self PrefixSuffix) Match(s string) bool { return strings.HasPrefix(s, self.Prefix) && strings.HasSuffix(s, self.Suffix) } func (self PrefixSuffix) String() string { return fmt.Sprintf("<prefix_suffix:[%s,%s]>", self.Prefix, self.Suffix) }

// Copyright (c) 2002-2020 "Neo4j," // Neo4j Sweden AB [http://neo4j.com] // // This file is part of Neo4j. // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. using System; using System.Collections.Generic; namespace Neo4j.Driver { /// <summary> /// Common interface for components that can execute Neo4j queries. /// </summary> /// <remarks> /// <see cref="IAsyncSession"/> and <see cref="IAsyncTransaction"/> /// </remarks> public interface IQueryRunner : IDisposable { /// <summary> /// /// Run a query and return a result stream. /// /// This method accepts a String representing a Cypher query which will be /// compiled into a query object that can be used to efficiently execute this /// query multiple times. /// </summary> /// <param name="query">A Cypher query.</param> /// <returns>A stream of result values and associated metadata.</returns> IResult Run(string query); /// <summary> /// Execute a query and return a result stream. /// </summary> /// <param name="query">A Cypher query.</param> /// <param name="parameters">A parameter dictionary which is made of prop.Name=prop.Value pairs would be created.</param> /// <returns>A stream of result values and associated metadata.</returns> IResult Run(string query, object parameters); /// <summary> /// /// Run a query and return a result stream. /// /// This method accepts a String representing a Cypher query which will be /// compiled into a query object that can be used to efficiently execute this /// query multiple times. This method optionally accepts a set of parameters /// which will be injected into the query object query by Neo4j. /// /// </summary> /// <param name="query">A Cypher query.</param> /// <param name="parameters">Input parameters for the query.</param> /// <returns>A stream of result values and associated metadata.</returns> IResult Run(string query, IDictionary<string, object> parameters); /// <summary> /// /// Execute a query and return a result stream. /// /// </summary> /// <param name="query">A Cypher query, <see cref="Query"/>.</param> /// <returns>A stream of result values and associated metadata.</returns> IResult Run(Query query); } }

Q: Why doesn't the IAU consider Quaoar to be a dwarf planet? Only Ceres, Pluto, Haumea, Makemake and Eris are listed by the IAU as dwarf planets. Quaoar is bigger than Ceres, yet it doesn't make the cut. In fact, it fits the IAU description of a dwarf planet The shape of objects with mass above $5 \times 10^{20}$ kg and diameter greater than 800 km would normally be determined by self-gravity since Quaoar has a mass of $1.4 \times 10^{21} $ kg and a diameter of 1110 km. Futhermore, light-curve-amplitude analysis shows that Quaoar is indeed a spheroid. Why isn't Quaoar considered to be a dwarf planet by the IAU ? A: It probably is a dwarf planet. (It almost certainly is a dwarf planet.) The naming procedures at the IAU are that "Objects that have an absolute magnitude (H) less than +1 [...] are overseen by two naming committees, one for minor planets and one for planets. [...] All other bodies are named by the minor-planet naming committee alone." source—wikipedia Quaoar has an absolute magnitude of +2.4, so it has its name approved by the minor planet committee, and that committee doesn't rule on whether it is a dwarf planet or not. It's just procedural. So it's not that this is a particularly controversial object: It's big and round enough to fit the criteria for a dwarf planet. It just doesn't have the stamp of approval from the planet naming committee, and they don't get out bed for anything with an absolute magnitude of more than +1. As observed in a comment, officially naming objects or putting an official box-label on a particular object may not get a particularly high priority in the scientific context. What an object is referred to is what gets established in debate and an "official" discussion and label is only be granted in particularly controversial cases of great general interest

function y = vl_nnnormalizelp(x,dzdy,varargin) %VL_NNNORMALIZELP CNN Lp normalization % Y = VL_NNNORMALIZELP(X) normalizes in Lp norm each spatial % location in the array X: % % Y(i,j,k) = X(i,j,k) / sum_q (X(i,j,q).^p + epsilon)^(1/p) % % DZDX = VL_NNNORMALIZELP(X, DZDY) computes the derivative of the % function with respect to X projected onto DZDY. % % VL_NNNORMALIZE(___, 'opts', val, ...) takes the following options: % % `p`:: 2 % The exponent of the Lp norm. Warning: currently only even % exponents are supported. % % `epsilon`:: 0.01 % The constant added to the sum of p-powers before taking the % 1/p square root (see the formula above). % % `spatial`:: `false` % If `true`, sum along the two spatial dimensions instead of % along the feature channels. % % See also: VL_NNNORMALIZE(). opts.epsilon = 1e-2 ; opts.p = 2 ; opts.spatial = false ; opts = vl_argparse(opts, varargin, 'nonrecursive') ; if ~opts.spatial massp = sum(x.^opts.p,3) + opts.epsilon ; else massp = sum(sum(x.^opts.p,1),2) + opts.epsilon ; end mass = massp.^(1/opts.p) ; y = bsxfun(@rdivide, x, mass) ; if nargin < 2 || isempty(dzdy) return ; else dzdy = bsxfun(@rdivide, dzdy, mass) ; if ~opts.spatial tmp = sum(dzdy .* x, 3) ; else tmp = sum(sum(dzdy .* x, 1),2); end y = dzdy - bsxfun(@times, tmp, bsxfun(@rdivide, x.^(opts.p-1), massp)) ; end

Background ========== Hereditary Angioedema (HAE) is an autosomal dominant disorder resulting from a deficiency of C1 esterase inhibitor (C1-INH). It is a rare disease with clinical manifestations debilitating and potentially fatal. The aim of this study was to report the clinical and laboratory characteristics and treatment of patients with Hereditary Angioedema with C1-INH deficit Outpatient Immunology University. Methods ======= This was a retrospective study using data from the clinical records of patients with confirmed HAE with C1-INH diagnosis. The laboratory diagnosis was made after dosages of C4 and C1-INH and functional study of C1-INH (Technoclone ® kit). Age at time of first appointment, onset of symptoms, time to diagnosis, clinical manifestations, prodrome, angioedema triggered the crisis, the need for hospitalization, prophylactic treatment and medication used for seizures were analyzed. Results ======= Were included 30 patients (22F:8M; 16 days of age -- 51 years old) diagnosed in the last 2 years. The first symptoms occurred: (6/30; 20%) before 2 years old; (6/30; 20%), most of cases (10/30; 33,3%) ocurred in the adolescence and two patients were asymptomatic. The following clinical manifestations were reported: subcutaneous edema in 86%; 56.6% affecting the face; abdominal pain in 80% and 33.3% of them were submited to abdominal surgery; 46.6% reported asphyxia and 28,5% had voice changing. Prodromal symptoms were referred in 36.6%: cutaneous rash, tingling and pruritus, triggering factors were: trauma (6/30; 53.3 %), stress (17/30; 56.6%) and pregnancy was reported by 4 patients. Hospitalization was referred by 63.3% and out of them, 21% in Intensive Care Unity (ICU). Therapy was employed: danazol (14/30), oxandrolon (13/30) and tranexamic acid (15/30), plasma (4/30). Icatibant was available and applied in 12 patients. Conclusions =========== Clinical manifestations did not differ from the previous reports. It was relevant the high frequency of hospitalization as well as ICU admission. This situation reflects the restricted access to therapy of the attacks. In addition, previous abdominal surgery was also reported in a third of the patients. Although the knowledge about HAE has improved in our country, the access to therapy and management as a whole are still restricted. \* Study supported with a grant from Shire Research Program.

Neopets Customisation What is Neopets Customisation? On April 26th 2007 Neopets launched their newest project along with a new site layout. The new and prestigious project was that you can dress up (customise) your pets. In order for people to be able to dress up their pets, all the pets had changed their look a little so they would be able to wear the clothes. The second form of clothing is the one that comes with paint brushes, for example a Royal Paint Brush will give your pet lovely medieval clothes to put on. The third form of clothing you can buy with real money at the NC Mall. Who can wear clothes? Pets that are not in a normal shape such as baby pets, fruit/vegetable chias, mutant and maraquan pets cannot wear clothes. Pets that are not converted cannot wear clothes either. All of these pets can wear backgrounds and trinkets. All other pets can wear clothes, backgrounds and trinkets. What is on this page? This page shows you an overview of all the clothes (from shops, not paint brushes) that are currently available for your pet to wear. Use the navigation below to get to the right page. If you want to see the paint brush clothes, visit Deluxe Customisation. If you want to see the NC Mall clothes, visit NC Mall Customisation. Help Us You can help us by modeling your pets wearing certain clothing items. If you'd like to help model, please visit our Model for SunnyNeo page to see which items we currently need modeled. Also, the items shown below have not yet been released by Neopets. If you see these items in shops, auctions, trading post or worn by a neopet, please send a neomail to crowprincess. A B C D E F G H I J K L M N O Also, please neomail crowprincess if you get any wearable items from redeeming Neocash Cards, from redeeming codes from the Neopets Puzzle Adventure game or from any other promotion.

Q: Update Position of ThreeJS model inside Animate (How to access model's position outside of callback) I am attempting to create a model of the ISS orbiting the earth. I am using threeJS to accomplish this. So far I have everything working perfectly other than I am having trouble with updating the position of the imported ISS model. I have a sphere that orbits the earth getting its position from an AJAX call every 5 seconds that is then converted to coordinates on my earth sphere model. I essentially want to replace the sphere (posSphere in code) that orbits, with the model of the ISS. Having trouble as the model is loaded asynchronously. What I have currently tried is including parameter in the animate function for the model, and on callback send it to the animate function, but this causes the entire scene to freeze. <!DOCTYPE html> <html lang="en"> <head> <title>three.js webgl - OBJLoader + MTLLoader</title> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0"> <style> body { font-family: Monospace; background-color: #000; color: #fff; margin: 0px; overflow: hidden; } </style> </head> <body> <script src="Resources/three.js"></script> <script src="Resources/TDSLoader.js"></script> <script src="Resources/FBXLoader.js"></script> <script src="Resources/GLTFLoader.js"></script> <script src="Resources/inflate.min.js"></script> <script src="Resources/TrackballControls.js"></script> <script> var lat, long, issPosition; //*********************PRELIM FUNCTIONS BEGIN********************************** //AJAX request for current position of the ISS function GetValue() { var xhr = new XMLHttpRequest(); xhr.onreadystatechange = function() { if (this.readyState == 4 && this.status == 200) { var requestResponse = xhr.responseText; var issInfo = JSON.parse(requestResponse); var Lat = issInfo.iss_position.latitude; var Long = issInfo.iss_position.longitude; callback(Lat, Long); //callback function with lat and long info } }; xhr.open("GET", "http://api.open-notify.org/iss-now.json", true); xhr.send(); } function callback(Lat, Long) { lat = Lat; //set global variables equal to lat and long so animate function has access long = Long; } GetValue(); //function call to get iss location setInterval(GetValue, 5000); //interval for iss location, updates every 5 seconds //convert long & lat to 3D coordinate function latLongToVector3(lat, lon, radius, heigth) { var phi = (lat)*Math.PI/180; var theta = (lon-180)*Math.PI/180; var x = -(radius+heigth) * Math.cos(phi) * Math.cos(theta); var y = (radius+heigth) * Math.sin(phi); var z = (radius+heigth) * Math.cos(phi) * Math.sin(theta); return new THREE.Vector3(x,y,z); } //******************PRELIM FUNCTIONS END******************************************** //******************THREE JS ENVIRONMENT BEGIN************************************** var width = window.innerWidth; var height = window.innerHeight; var scene = new THREE.Scene(); var camera = new THREE.PerspectiveCamera(75, width/height, 0.01, 1000); camera.position.z = 400; var controls = new THREE.TrackballControls( camera ); controls.rotateSpeed = 1.0; controls.zoomSpeed = 1.2; controls.panSpeed = 0.8; controls.noZoom = false; controls.noPan = false; controls.staticMoving = true; controls.dynamicDampingFactor = 0.3; controls.keys = [ 65, 83, 68 ]; var renderer = new THREE.WebGLRenderer(); renderer.shadowMap.enabled = true; renderer.shadowMap.type = THREE.PCFSoftShadowMap; renderer.setSize(width, height); document.body.appendChild(renderer.domElement); var direcLight = new THREE.DirectionalLight(0xffffff, 0.3); direcLight.position.set(-3,3,1.5); direcLight.castShadow = true; scene.add(direcLight); var ambientLight = new THREE.AmbientLight(0xc9c9c9, 1.5); scene.add(ambientLight); var geometry1 = new THREE.SphereGeometry(200,32,32); var geometry2 = new THREE.SphereGeometry(202.5,32,32); var geometry3 = new THREE.SphereGeometry(3, 32, 32); var material1 = new THREE.MeshBasicMaterial ({ color: 0xff0000 }); var material2 = new THREE.MeshPhongMaterial({ map: new THREE.TextureLoader().load('Resources/Earth3/earthmapoftwo.jpg'), bumpMap: new THREE.TextureLoader().load('Resources/Earth3/Bump2.jpg'), bumpScale: 1, specularMap: new THREE.TextureLoader().load('Resources/Earth3/oceanmaskbytwo.png'), specular: new THREE.Color('grey'), shininess: 40 }); var material3 = new THREE.MeshPhongMaterial({ alphaMap: new THREE.TextureLoader().load('Resources/Earth3/Clouds.png'), transparent: true, }); var issModel; var loaderGLTF = new THREE.GLTFLoader(); loaderGLTF.load( "Resources/Earth3/iss.gltf", function(gltf) { //I want to be able to access this gltf outside of this callback function gltf.scene.scale.x = 0.1; gltf.scene.scale.y = 0.1; gltf.scene.scale.z = 0.1; console.log(gltf.scene.position); scene.add(gltf.scene); animate(gltf); //send gltf to animate function, this does not work } ) var sphere = new THREE.Mesh(geometry1, material2); var clouds = new THREE.Mesh(geometry2, material3); var posSphere = new THREE.Mesh(geometry3, material1); sphere.receiveShadow = true; clouds.receiveShadow = true; posSphere.receiveShadow = true; scene.add(sphere); scene.add(clouds); scene.add(posSphere); function animate(gltf) { clouds.rotation.x += 0.0001; clouds.rotation.y += 0.0001; clouds.rotation.z += 0.0001; issPosition = latLongToVector3(lat, long, 200, 10); posSphere.position.x = issPosition.x; //sphere's location is updated with the issPosition & works posSphere.position.y = issPosition.y; posSphere.position.z = issPosition.z; gltf.scene.position.x = issPosition.x; //Part I am struggling with here, how to access the gltf's position from within this function. Essentially I want to replace the sphere with the gltf gltf.scene.position.y = issPosition.y; gltf.scene.position.z = issPosition.z; controls.update(); requestAnimationFrame(animate); renderer.render(scene, camera); } animate(); //**************************THREE JS ENVIRONMENT END************************************* </script> </body> </html> A: You're defining a parameter to the animate function but then immediately calling it with no parameter.. (down at the last line of your code). that's why your app is "freezing". (its not freezing.. it's probably throwing an exception.. You can see that in the debugger console.. Ctrl+Shift+J in chrome or More Tools->Developer Tools from the ... menu in the upper right.. Exceptions should show up in red at the bottom.) Take out the line of code where you do 'animate(gltf)' and instead do "issModel = gltf.scene;" Then in your animate function instead of doing "gltf.scene.position.x = " etc. Just do "issModel.position.copy(issPosition);" If you're still stuck, try hosting it online somewhere and we can take a deeper look.

Filipino art and literature The Ifugao tribe made iconic sculptures Featuring a rich blend of elements from Malaysia, Spain, America and a whole host of other nations, the artistic culture of the Philippines is richly diverse. Many of the traditional tribes that can be found in remote areas of the country and on some of the smaller islands also have their own unique practices, that all fall into the melting pot that is the Philippines. Artistic expression in the Philippines The Filipino people are naturally very creative, and art forms an important part of the culture here. Many people seem to be able to pick up a brush almost at will and create stunning works of art, while other people are skilled at cooking, woodcarving and other skills such as weaving. Northern Luzon is full of fascinating tribes such as the Ifugao people who built the amazing rice terraces around Banaue around 3,000 years ago. One of the best ways to discover the full range of artistic expression in the Philippines is during a festival, when people gather to share their skills and create colourful costumes as well as floats, special dishes and a whole host of other items. There are countless art galleries to be found in the Philippines, and these are good places to discover the way that the artwork here has developed over the centuries. However, artwork in all forms can be found all over the country in various venues, from pieces that are proudly displayed in homes and businesses to those on market stalls and at the side of the street. Travel fact The oldest surviving written Filipino text was discovered near Laguna de Bay and has been preserved on copperplate. Dating back to around 900AD, this unique text is a record of payment of a debt using 856g of gold, and the language used resembles Old Javanese and Sanskrit. Literature and theatre in the Philippines Literature has played an important role in the liberation of Filipino people and the patriotic novels of Philippine national hero Jose Rizal are said by many people to have paved the way to freedom from the oppression of the Spaniards. In the past, legends and myths were passed down by word of mouth, and it wasn’t until the 17th century that these stories started to be written down. Original stories and poems also started to emerge at this time, and while the oldest of these were written in Tagalog, many of the more modern tales tend to be written in English. Theatre has long played an important role in the culture of the Philippines, and the oldest of the traditional plays tend to depict the life of Jesus Christ. Known as cenaculo, these plays are extremely popular, while the moro-moro plays are comical in nature and present the feuds between the Christian and the Muslim people in the Philippines. Adapted from Spanish origins, the zarzuela plays are also popular and are a type of operetta. There is also a thriving modern theatre industry in the Philippines, which tends to take its inspiration from popular Broadway productions as well as avant-garde stage shows. Those who wish to gain an insight into the theatrical traditions of the Philippines should pay a visit to one of the country’s main theatrical institutions such as the Cultural Centre of the Philippines, which proudly displays the nation’s theatrical skills. Visitors will also want to witness Balagtasan, which is an interesting type of spontaneous poetic debate that is performed by poets who compete against each other by reciting spontaneous poetic verses to demonstrate their points of argument.

Q: How to handle a custom date format coming from json data with angularjs? I have a angularjs service: angular.module('myApp.services', ['ngResource']) .factory('Scheduler', function ($resource) { return $resource('/api/scheduler/:id', {id:'@id'}, { 'get' :{method:'GET', isArray:false}, 'save' :{method:'PUT'} }); }) which returns the following json: { 'start': '08:00:00', 'end': '21:30:00', 'offset': 5 } Is there e preferred way in angularjs to convert the 'start date' and the 'end date' in a js Date Object and back to a string right before 'save' it back to the webservice? Some kind of function which is called right before? A: It it not supported by the $resource service at the moment, use the $http instead. With $http you can have request and response transformers.

/// Helper for 'components/category' to stylize a category with a specified color /// @param {Color} $color - Color CSS value @mixin category-color-variant($color) { color: $color !important; border: 1px solid $color; &:hover { color: darken($color, 15) !important; border: 1px solid darken($color, 15); text-decoration: none; } } /// Helper for 'components/category' to stylize a category with a specified size and padding /// @param {Number} $font-size - Font-size CSS value /// @param {List} $padding - Padding CSS value @mixin category-size-variant($font-size, $padding) { font-size: $font-size; padding: $padding; }

All relevant data are within the paper and its Supporting Information files. Introduction {#sec001} ============ The cellular response to stress induces expression of chaperone proteins such as heat shock protein 70 (Hsp70). These chaperones help to refold proteins and also work with ubiquitination machinery to degrade proteins that cannot be refolded \[[@pone.0128240.ref001]\]. When the crisis is over, the induced proteins must then be degraded to restore the normal cellular activities. Hsc70 is the constitutive chaperone homolog of Hsp70 that serves in a similar role to assist in protein refolding under normal cellular conditions. The significance of proper maintenance of the balance of constitutive and stress-induced chaperones such as Hsc70 and Hsp70 is underscored by alterations of the levels and clearance rate of pathogenic proteins like the neurodegeneration-associated tau when this balance is misregulated \[[@pone.0128240.ref002]\]. The heat shock protein co-chaperone CHIP functions with Hsp70 and Hsc70 in protein triage \[[@pone.0128240.ref003]\]. In addition to assisting in the refolding of substrates, CHIP is an E3 ubiquitin ligase that catalyzes the polyubiquitination of substrates, which leads to their degradation in the 26S proteasome. CHIP also plays a role in maintaining the cellular balance of Hsp70 and Hsc70 as it will polyubiquitinate these chaperones when they carry no client \[[@pone.0128240.ref004],[@pone.0128240.ref005]\]. Studies in vitro and in cells have shown that whereas both Hsp70 and Hsc70 are ubiquitinated by CHIP, only the inducible form (Hsp70) is substantially degraded \[[@pone.0128240.ref006]\]. This is logical from a functional perspective as the inducible form must be removed when cell stress is alleviated whereas levels of the constitutive form need to remain constant. However, degradation of only Hsp70 is surprising because the two proteins share \>86% sequence identity and 93% sequence conservation. This poses a significant challenge to our understanding of both the subtleties of polyubiquitin chain formation and the recognition of specific chain types as a signal for degradation. Ubiquitination of proteins involves the action of a cascade of E1 activating, E2 conjugating and E3 ligating enzymes. Ubiquitination can signal for an array of different responses, which are determined by the extent (number) and type (chain linkage and forking of chains) of ubiquitin (Ub) molecules transferred to the substrate \[[@pone.0128240.ref007]\]. As with other post-translational modifications, the position of the modification on the substrate is a key, although not the sole, factor determining the functional outcome. CHIP has been shown to produce all known types of ubiquitination modifications including K48-linked and forked Ub chains \[[@pone.0128240.ref008]\]. A previous study examined CHIP-mediated ubiquitination of Hsp70 and reported six sites that were ubiquitinated based on proteomic analysis with \~64% coverage of the protein \[[@pone.0128240.ref009]\]. Here, we report a more in-depth and systematic biochemical and proteomic analysis of CHIP ubiquitination of Hsp70 and also compare the results to the corresponding analysis of the highly similar Hsc70 protein. These studies represent one of the most in-depth investigations of in vitro ubiquitination reported to date and provide clear evidence of differences in the ubiquitination of these two homologous chaperones. Materials and Methods {#sec002} ===================== Protein purification {#sec003} -------------------- All proteins were purified from recombinant expression in *E*. *coli* and correspond to human sequences for each protein except for E1, which is the wheat Uba1. E1 (Uba1), UbcH5a, Ube2W, UbcH13, Uev1a, CHIP, Hsp70, and WT Ub were purified as before except that for some experiments the affinity tag of Hsp70 was not cleaved \[[@pone.0128240.ref010]\]. The plasmid expressing human Hsc70 was a gift from Jason Young (McGill University), and His-Hsc70 was purified using Ni-affinity and size exclusion chromatography. Plasmids encoding Ub mutants K0, K6R, K48R and K63R were a gift from Rachel Klevit (University of Washington) and His-Ub K11R was a gift from Michael Rape (University of California, Berkeley). The untagged Ub mutants were purified by lysing cells in buffer containing 30 mM ammonium acetate pH 5.1, then treating the cell lysate with acetic acid to lower the pH 4.8 and centrifuging to remove the precipitate. The protein was then further purified using cation exchange with ammonium acetate buffer and then size exclusion chromatography. All proteins were stored at -80°C in buffer containing 20 mM Tris pH 7.5, 1 mM DTT and either 50 mM NaCl (E1 and Ub) or 100 mM NaCl (E2s, CHIP, Hsc70, and Hsp70). In Vitro Ubiquitination {#sec004} ----------------------- CHIP autoubiquitination was performed as previously described with the time of incubation for each sample as noted \[[@pone.0128240.ref010]\]. Substrate ubiquitination for Hsc70 and Hsp70 was optimized for a minimum concentration of CHIP to reduce auto-ubiquitination species and performed in a manner similar to the autoubiquitination reactions with 1.5 μM Hsc70 or Hsp70, 0.15 μM CHIP, and 40 μM Ub. Briefly, all reactions were incubated in buffer containing 100 mM NaCl, 40 mM Tris pH 7.5, 5 mM MgCl~2~, and 5 mM ATP at 30°C for the times indicated. The reaction products were then separated by SDS-PAGE with 4--12% gradient gels and either stained with Coomassie Blue or transferred to PVDF membrane for a western blot. Immunoblotting used either 1:3000 monoclonal mouse anti-Ub (Abcam), 1:3000 polyclonal rabbit anti-CHIP (Calbiochem), 1:2000 monoclonal mouse anti-Hsp70 (Enzo Life Sciences), or 1:3000 monoclonal mouse anti-GST (GenScript) and the appropriate HRP-fused secondary antibodies at 1:7500--10000. Proteasomal degradation {#sec005} ----------------------- In vitro degradation was allowed to proceed in concert with ubiquitination. The reactions were altered to better match the previous work of Qian et al 2006 such that conditions included 100 nM E1, 2 μM E2, 3 μM CHIP, 1 μM Hsx70, 50 μM Ub, in 30 mM HEPES pH 7.5, 20 mM NaCl, 1 mM DTT, 5 mM MgCl~2~, 5 mM ATP with a creatine phosphate ATP-regeneration system. Reactions were incubated at 37°C for up to 2 hours with 45 nM 26S proteosome as indicated and the products analyzed as above for in vitro ubiquitination. Mass spectrometry proteomics {#sec006} ---------------------------- Following SDS-PAGE analyses, gels were stained with Coomassie stain, and Hsc70 and Hsp70 bands were excised and cut into 1 mm^3^ pieces. These gel pieces were first treated with 45 mM DTT for 20 minutes, followed by treatment with 100 mM iodoacetamide for 20 minutes. After destaining with 50% MeCN in 50 mM ammonium bicarbonate, the gel bands were digested with sequencing-grade trypsin in 25 mM ammonium bicarbonate overnight at 37°C. Peptides were extracted by gel dehydration (60% MeCN, 0.1% TFA), the extracts were dried by vacuum centrifugation, and peptides were reconstituted in 0.1% formic acid. Extracted peptides were then separated by reverse phase liquid chromatography and analyzed by tandem mass spectrometry (MS/MS). For LC-MS/MS analysis, the peptide extracts were first loaded onto a capillary reverse phase analytical column (360 μm O.D. x 100 μm I.D.) using an Eksigent NanoLC Ultra HPLC and autosampler. The analytical column was packed with 20 cm of C18 reverse phase material (Jupiter, 3 μm beads, 300 Å, Phenomenox), directly into a laser-pulled emitter tip. Peptides were gradient-eluted at a flow rate of 500 nL/min, and the mobile phase solvents consisted of 0.1% formic acid, 99.9% water (solvent A) and 0.1% formic acid, 99.9% acetonitrile (solvent B). A 90-minute gradient was performed, consisting of the following: 0--10 min, 2% B; 10--50 min, 2--40% B; 50--60 min, 40--95% B; 60--65 min, 95% B; 65--70 min 95--2% B; 70--90 min, 2% B. Eluting peptides were mass analyzed on an LTQ Orbitrap XL or an LTQ Orbitrap Velos mass spectrometer (Thermo Scientific), each equipped with a nanoelectrospray ionization source. For the majority of LC-MS/MS analyses, the instruments were operated using a data-dependent method with dynamic exclusion enabled. Full scan (m/z 300--2000 or 400--2000) spectra were acquired with the Orbitrap (resolution 60,000). For LTQ Orbitrap XL analyses, the top 5 most abundant ions in each MS scan were selected for fragmentation via collision-induced dissociation (CID) in the LTQ ion trap. A data-dependent method involving selection of the top 16 most abundant ions per MS scan was used for LTQ Orbitrap Velos analyses. An isolation width of 2 m/z, activation time of 10 or 30 ms, and 35% normalized collision energy were used to generate MS2 spectra. Dynamic exclusion settings allowed for a repeat count of 2 within a repeat duration of 10 sec, and the exclusion duration time was set to 15 sec. For selected LC-MS/MS analyses, the LTQ Orbitrap Velos was operated using a method consisting of data-dependent and targeted scan events, for which specific *m/z* values corresponding to Hsp70 or Hsc70 ubiquitinated peptides of interest were provided in the data acquisition method to facilitate targeted MS/MS spectra despite the intensity of peptide precursors. For identification of Hsp70 and Hsc70 peptides, tandem mass spectra were searched with Sequest \[[@pone.0128240.ref011]\] against an *E*. *coli* subset database created from the UniprotKB protein database ([www.uniprot.org](http://www.uniprot.org/)), appended with protein sequences for the recombinantly expressed proteins E1, UbcH5a, Ube2W, UbcH13, Uev1a, CHIP, Hsc70, Hsp70 and Ub depending on their presence in the sample. Variable modifications of +57.0214 on Cys (carbamidomethylation), +15.9949 on Met (oxidation), and +114.0429 on Lys residues (corresponding to the diGly remnant that remains after tryptic cleavage of ubiquitin) were included for database searching. Search results were assembled using Scaffold 3.6.4 (Proteome Software). Spectra acquired of Hsc70 and Hsp70 peptides were then inspected using Xcalibur 2.1 Qual Browser software (Thermo Scientific). Tandem mass spectra of ubiquitinated peptides as well as spectra acquired of the corresponding unmodified peptide forms were examined by manual interrogation, and sites of ubiquitination on modified peptides were validated. Structural models were created based on the ADP-bound form of the *E*. *coli* homolog DnaK (PDBID 2KHO\[[@pone.0128240.ref012]\]) using the SWISS-MODEL web server\[[@pone.0128240.ref013]\]. Results {#sec007} ======= The objective of this study was to systematically characterize the ubiquitination of Hsp70 by the E3 ubiquitin ligase CHIP and identify any differences relative to ubiquitination of Hsc70. Our laboratory previously established that CHIP could produce polyubiquitin chains only with either an E2 from the UbcH5 family or the combination of a mono-ubiquitinating E2 (Ube2W or UbcH6) with the chain-building E2 UbcH13/Uev1 \[[@pone.0128240.ref010]\]. Our studies focused on the analysis of UbcH5-mediated ubiquitination because it generates a diversity of products that could lead to differences in their rates of proteasomal degradation. Nevertheless, we also analyzed CHIP ubiquitination with the Ube2W E2 enzyme to determine if the modification of the heat shock proteins was the same as for CHIP ubiquitination with UbcH5. Importantly, a link between in vitro results and in vivo (cellular) degradation has been made in the previous study of CHIP-mediated ubiquitination of Hsx70 proteins \[[@pone.0128240.ref006]\], albeit with a greater emphasis on degradation relative to our focus on ubiquitination. In particular, that study showed that Hsp70 was degraded faster and more completely than Hsc70 in both in vitro assays using purified components and in mouse fibroblast cells. While, our goal was to follow up on the initial observation and obtain detailed knowledge of CHIP-mediated ubiquitination of the Hsx70 proteins, to ensure the two studies were aligned, we confirmed that the ubiquitinated protein produced by our assays follows the same trend of degradation as reported previously (*vide infra*). Identification of ubiquitination sites by tandem mass spectrometry {#sec008} ------------------------------------------------------------------ Hsp70 and Hsc70 have a high degree of sequence identity (86%) and most lysines are conserved ([S1 Fig](#pone.0128240.s001){ref-type="supplementary-material"}). While there is one reported analysis of human Hsp70 ubiquitination by CHIP with UbcH5, no data are available for Hsc70. We therefore performed a comparative proteomic analysis to extend the analysis of CHIP ubiquitination of Hsp70 and identify the differences with respect to ubiquitination of Hsc70. The in vitro ubiquitination reactions were optimized to obtain the highest yields and resolution of the ubiquitination ladder products under multiple turnover conditions ([Fig 1A](#pone.0128240.g001){ref-type="fig"}). To support the correlation of the in vitro data with the degradation in vivo, as has been established previously, we examined the degradation of ubiquitinated Hsp70 relative to ubiquitinated Hsc70 in experiments with purified 26S proteasome added ([Fig 2](#pone.0128240.g002){ref-type="fig"}). While these data do not exactly replicate the previously reported results, they show that the CHIP-ubiquitinated proteins produced in our assays exhibit the same trend of preferential degradation of Hsp70 versus Hsc70 in vitro and in vivo \[[@pone.0128240.ref006]\]. Thus, the in vitro ubiquitinated proteins are distinguishable in some manner, which is investigated below. {#pone.0128240.g001} {#pone.0128240.g002} Our analysis focused on the +1 Ub and +2 Ub bands, which were excised from the gel and trypsin digested, then the resultant peptides were extracted for LC-MS/MS analysis. Trypsin cleavage of a Ub modified lysine leaves a signature diGly tag that can be recognized as a post-translational modification of +114 Da. The samples from the +1 Ub bands are expected to contain a mixture of many different mono-ubiquitinated species representing the range of ubiquitination sites that are produced, while the +2 Ub bands will contain molecules with either two separate ubiquitination sites or a di-Ub addition to a single site. Examples of tandem mass spectra of ubiquitinated Hsx70 (both Hsp70 and Hsc70) peptides are shown in [Fig 1B and 1C](#pone.0128240.g001){ref-type="fig"}. Optimization of the proteomic analysis resulted in identification of peptides representing 84% of the sequence of Hsp70 compared to the 64% coverage in the previous study ([Table 1](#pone.0128240.t001){ref-type="table"}, [S1 Fig](#pone.0128240.s001){ref-type="supplementary-material"}). This enabled 39 of the 50 lysines in Hsp70 to be monitored for modification. Most of the lysine residues that could not be analyzed are located in regions of the proteins that when trypsin digested result in very short peptides, and therefore are not detectable in our LC-MS/MS experiments. Remarkably, while we anticipated that ubiquitination would be quite promiscuous, only 12 lysines in Hsp70 were modified at detectable levels. This included all 6 sites previously reported for Hsp70 \[[@pone.0128240.ref009]\] and an additional 6 sites in the new regions of the protein covered by our analysis. The di-ubiquitinated Hsp70 sample was specifically analyzed to look for Ub-Ub bonds and only K48-linked chains were found. This is in contrast to the four types of Ub linkages reported previously \[[@pone.0128240.ref009]\]. While the mass spectrometry experiments are not quantitative, predominance of K48-linked ubiquitin chains on Hsp70 would be consistent with degradation of the ubiquitinated protein. 10.1371/journal.pone.0128240.t001 ###### Summary of results from mass spectrometry proteomics analysis of +1 or +2 Ub samples from [Fig 1A](#pone.0128240.g001){ref-type="fig"}. {#pone.0128240.t001g} Sequence coverage Lys observed Ub sites --------------- ------------------- -------------- ---------- Hsc70 89% 45/54 16 Hsp70 84% 39/50 12 Ub (from Hsc) 92% 6/7 3 Ub (from Hsp) 92% 6/7 1 (K48) CHIP 40% 7/20 1 (K22) Proteomic analysis of Hsc70 was also completed with 89% of the sequence observed. For this protein 45 of the 54 lysines could be monitored, but again only a subset was detected with Ub modification. These 16 sites include 12 sites that are either identical to or within three residues of the Hsp70 ubiquitination sites ([S1 Fig](#pone.0128240.s001){ref-type="supplementary-material"}). Among the four lysine residues uniquely ubiquitinated in Hsc70, three involved sites for which there is no analogous lysine in Hsp70 (K458, K531, and K601). The fourth site, K159, is a lysine in Hsp70 and therefore represents the only direct difference in the ubiquitination sites for the Hsx70 proteins. To validate that this site is not ubiquitinated in Hsp70, a tandem mass spectrometry experiment was performed to target the predicted peptide ion of the K159-ubiquitinated Hsp70 peptide. Only a very weak signal could be observed for the expected mass, and the peptide could not be identified with confidence. Hence, if this site is modified at all it is at a very low level. The di-ubiquitinated Hsc70 was also specifically analyzed to look for Ub-Ub bonds and three types of linkages were observed for Hsc70 (K6, K11 and K48). It is important to note that these experiments are not quantitative in nature so firm conclusions cannot be drawn. Nevertheless, the results suggest that K48-linked Ub is the predominant product of ubiquitination of Hsp70 by CHIP, but not necessarily for Hsc70. The high overall coverage of the proteins in the proteomic analysis enabled us to investigate if a structural basis for the limited number of ubiquitination sites observed is evident. Mapping the ubiquitination results onto homology models of Hsc70 and Hsp70 reveals a broad distribution of ubiquitination sites with a slight concentration of modified sites in the C-terminal half of the proteins ([Fig 3](#pone.0128240.g003){ref-type="fig"}). This graphical analysis demonstrates that ubiquitination is not localized to a single face of either protein. CHIP binds to the dynamically disordered C-termini of Hsc70 and Hsp70 (not shown in [Fig 3](#pone.0128240.g003){ref-type="fig"}) and both Hsx70 proteins are known to be structurally dynamic with well-characterized conformational changes over the course of the ATP cycle. Hence, it is likely that a wide range of orientations of CHIP and the substrate are sampled during the assay, which should allow for ubiquitination of nearly any exposed Hsx70 lysine residue. {#pone.0128240.g003} Analysis of Hsx70 polyubiquitin linkage using Ub point mutations {#sec009} ---------------------------------------------------------------- To obtain further information on the Ub chain linkages formed on Hsc70 and Hsp70, ubiquitination reactions were performed using four single lysine to arginine Ub mutants (K6R, K11R, K48R, K63R) and Ub K0, which has all seven lysines mutated to arginine and cannot form lysine-linked chains. The loss of a single Lys site on Ub will restrict ubiquitination if that is the preferred site, but often will not entirely block activity as other sites can be used effectively under the in vitro conditions used. In order to perform studies of Hsp70 and Hsc70 ubiquitination, it is necessary to determine if the Ub mutants have a fundamental effect on CHIP activity even though all are readily conjugated to UbcH5a. To this end we performed CHIP auto-ubiquitination assays, which we had characterized extensively for WT Ub in a previous study \[[@pone.0128240.ref010]\]. We note that CHIP exhibits very robust auto-ubiquitination activity resulting in an amount of polyubiquitinated CHIP molecules that can be readily identified by Western blotting. [Fig 4A](#pone.0128240.g004){ref-type="fig"} compares results from CHIP autoubiquitination reactions with WT Ub and each of the 5 mutants. K11R Ub is a fusion protein with an N-terminal His tag and as such is a slightly larger protein and produces wider spacing in the Ub ladder. As expected, K0 Ub greatly reduced ubiquitination and bands are observed for only one or two Ub molecules added. The +2 Ub band presumably arises from addition to two different sites on the substrate. The products of Ub K11R, K48R and K63R reactions are all similar to WT Ub. The only mutant that did not follow expectations was K6R, for which the reactions seemed to produce less ubiquitinated product overall, particularly in the amount of highly ubiquitinated species. Together, these results provide the essential background for analyzing the effect of Ub mutations on CHIP substrate ubiquitination. {#pone.0128240.g004} CHIP auto-ubiquitination is very robust. In order to reduce the background in substrate ubiquitination reactions, the concentration of CHIP was reduced to the catalytic minimum. Despite this precaution, Western blots with substrate, CHIP and Ub antibodies of Hsx70 reactions reveal that most if not all of the high molecular weight species observed in Coomassie stained gels correspond to polyubiquitinated CHIP molecules ([S2 Fig](#pone.0128240.s002){ref-type="supplementary-material"}). Hence, we find that ubiquitination of the Hsx70 substrates by CHIP in vitro results in the addition of only a few (1--4) Ub molecules and little if any evidence of extensive polyubiquitination when the E3 is at catalytic concentrations. Hsc70 ubiquitination by CHIP appears to respond to Ub mutations in the same way as CHIP autoubiquitination. As expected, the Hsc70 ubiquitination assays show K0 Ub dramatically reduces the extent of substrate ubiquitination compared to WT Ub ([Fig 4B](#pone.0128240.g004){ref-type="fig"}). The proteomics based identification of Ub chain linkages predicts that Hsc70 ubiquitination with CHIP and UbcH5a, for which K6, K11, and K48 chains were observed, should be perturbed by the corresponding Ub K-R mutants. However, of these three Ub mutants only K6R appears to perturb the reaction, and this may be due to the reduction in CHIP activity with K6R Ub. In addition, we were surprised to observe that the K63R Ub mutant perturbed Hsc70 ubiquitination because we had not observed K63-linked chains in the proteomics study. Hsp70 ubiquitination showed much greater variation with the Ub mutants ([Fig 4C](#pone.0128240.g004){ref-type="fig"}). Again, K0 Ub allows only monoubiquitination of Hsp70. The proteomics based identification of Ub chain linkages predicts that Hsp70 ubiquitination should be perturbed by the Ub K48R mutant; however, K48R Ub does not appear to alter the extent of ubiquitination while K11R Ub does. Moreover, the reactions for both K6R and K63R Ub were also perturbed, albeit to a much lesser extent than K11R. To obtain further insight, the assays with Ub mutants were repeated with the addition of26S proteasome using the standard approach of monitoring the amount of intact, unmodified substrate ([S3 Fig](#pone.0128240.s003){ref-type="supplementary-material"}). Importantly, although only coarse estimates of degradation can be made, as noted above for WT Ub more robust degradation is observed for Hsp70 than Hsc70. As a consequence, we find little if any variability in the degradation of Hsc70 with the different mutants. In contrast, differences in the extent of degradation of Hsp70 for the different Ub mutants are detected. Confidence in the analysis is raised because the results appear to parallel the effect of Ub mutants in the assays where total ubiquitination is monitored ([Fig 4B and 4C](#pone.0128240.g004){ref-type="fig"}). Consistent with the very small amount of substrate ubiquitination observed in the absence of proteasome, the results for K0 and K6R Ub showed little evidence of degradation. For the K11R, K48R, and K63R Ub mutants, ubiquitination activity similar to WT was observed and degradation of Hsp70 for these mutants was similar to WT. Hsx70 ubiquitination with the Ube2W and UbcH13/Uev1 E2 conjugating enzymes {#sec010} -------------------------------------------------------------------------- Combined with our results from UbcH5a ubiquitination we investigated if the substrate specificity of CHIP depends on the identity of the E2. Moreover, analysis of ubiquitination by the combination of Ube2W and UbcH13/Uev1 E2 conjugating enzymes (W/13) provides further insight into ubiquitination by CHIP because it separates the initial mono-ubiquitination of substrates by Ube2W from polyubiquitination through K63-linked chains by UbcH13/Uev1. We first examined CHIP autoubiquitination for W/13 and compared the results to the products from reactions using UbcH5a ([Fig 4A](#pone.0128240.g004){ref-type="fig"}). As expected, since the W/13 E2 combination produces K63 chains, use of K0 or K63R Ub results in substantially reduced ubiquitination products and K11R reactions were similar to the WT control. Surprisingly, both K6R and K48R exhibited a marked decrease in the extent of ubiquitination such that no very high molecular weight species were produced. One possible explanation for these observations is that critical non-covalent interactions of Ub with Uev1a require the surface including K6 and K48, and that without them the chain extension activity of UbcH13/Uev1a is diminished\[[@pone.0128240.ref014]\]. We next turned to analysis of CHIP ubiquitination of Hsc70 with W/13 ([Fig 4B](#pone.0128240.g004){ref-type="fig"}). WT Ub produces a faint Ub ladder. This is reduced to monoubiquitination with both K0 and K63R Ub as neither can produce chains with W/13. The other three Ub mutants all have a ladder of products that resemble the extent of ubiquitination with WT Ub. In contrast, the ubiquitination reactions for Hsp70 with W/13 have much lower levels of ubiquitination overall, with only +1 and +2 Ub bands prominent even for WT Ub. As expected, K0 and K63R Ub are only capable of monoubiquitination, and K6R and K48R Ub products are similar to WT Ub. K11R Ub generates much less reaction, as was observed with the E2 enzyme UbcH5a, which suggests a general role of K11 in CHIP ubiquitination of Hsp70. A surprising observation was made in the Hsx70 ubiquitination reactions with the E2 enzymes W/13 and the K6R and K48R Ub mutants. The Ub ladders produced from these reactions included not only the normal ladder of +1, +2, +3 Ub but also produced extra bands in between the expected bands. To obtain further insight, we examined reactions with Ube2W alone to isolate the monoubiquitination function ([S4 Fig](#pone.0128240.s004){ref-type="supplementary-material"}). Initial mass spectrometry proteomics analysis of the band at the +1.5 Ub region on the gel from the reaction with Hsp70, W/13, and K48R Ub ([S4 Fig](#pone.0128240.s004){ref-type="supplementary-material"}, lane 13) did not identify any lysine ubiquitination sites on Hsp70 despite good coverage by mass spectrometry ([S5 Fig](#pone.0128240.s005){ref-type="supplementary-material"}). Previous studies have strongly implied that Ube2W is only capable of catalyzing monoubiquitination \[[@pone.0128240.ref010],[@pone.0128240.ref015]\]. Moreover, recent reports have shown that Ube2W will predominantly attach Ub to the N-terminus of substrates\[[@pone.0128240.ref016],[@pone.0128240.ref017]\]. Indeed, when searches were limited to ubiquitination of lysine residues, no modifications were observed. After extending the search, the proteomic analysis of Hsp70-Ub bands produced with Ube2W and WT, K0 and K48R Ub (bands from [S5 Fig](#pone.0128240.s005){ref-type="supplementary-material"} lanes 3, 4, and 7), all showed Hsp70 is predominantly modified on the N-terminus. In fact, the signal for this linear peptide is approximately 100-fold stronger than other ubiquitinated peptides we have seen and the signature diGly addition can be clearly identified in the y-ion series ([S6 Fig](#pone.0128240.s006){ref-type="supplementary-material"}). In addition to the primary site N-terminal ubiquitination with Ube2W, experiments with WT Ub revealed two internal Hsp70 lysines were ubiquitinated, K507 and K524. These signals were weak and required targeted analysis to obtain sufficient signal intensity to confirm the peptide assignment. Although not quantitative, these observations imply the modifications occur at much lower levels than the N-terminal modification. Importantly, the secondary ubiquitination sites provide an explanation for the +2, etc bands in reactions with WT and K11R Ub since there are multiple acceptor sites. The single +1 Ub band with the other Ub mutants (e.g. K0) and lack of identifiable internal lysine sites by proteomics both suggest there is less internal ubiquitination in reactions with these mutants. To further investigate the products of Ube2W reactions and determine the origin of the +1.5 Ub bands, we performed experiments using the method of Tatham et al.\[[@pone.0128240.ref016]\] in which the Hsp70 substrate is fused to a cleavable tag to enable N-terminal ubiquitination to be distinguished from internal lysine ubiquitination of the substrate. Ubiquitination products from reactions with the His-GST-Hsp70 fusion protein were treated to cleave the fusion protein, then all products were separated by SDS-PAGE ([Fig 5A and 5B](#pone.0128240.g005){ref-type="fig"}). Hsx70 and GST antibodies were used to confirm the identity of cleaved fragments ([Fig 5C and 5D](#pone.0128240.g005){ref-type="fig"}). As expected, the majority of the ubiquitinated species for all reactions containing Ube2W were associated with the N-terminal His-GST fragment with little ubiquitination of Hsp70 itself. In reactions with UbcH5a, some Hsp70-Ub was detected confirming the observation of internal lysine ubiquitination ([Fig 5C](#pone.0128240.g005){ref-type="fig"}; lane 3). Hsp70-Ub was also observed in reactions with Ube2W and WT Ub albeit at much lower levels ([Fig 5B](#pone.0128240.g005){ref-type="fig"}; lane 4). {#pone.0128240.g005} These results prompted a further investigation of the origin of the +1.5 Ub band from the K48R Ub reaction with W/13 ([S4 Fig](#pone.0128240.s004){ref-type="supplementary-material"}, lane 13). To extend the proteomics analysis, a targeted search was performed for the internal lysine sites previously found with Ube2W and WT Ub (K507 and K524), but only the N-terminal ubiquitination could be confirmed. Interestingly, this analysis revealed the presence of K63-linked di-Ub, which indicates that at least some of the species in this band have linked Ub molecules. In the more sensitive biochemical experiments with the cleavable His-GST-Hsp70 fusion protein, the +1.5 Ub intermediate bands are seen for K6R and K48R Ub in W/13 reactions, and following cleavage of the protein the +1.5 Ub ubiquitination maps to the N-terminal GST fragment rather than the Hsp70 protein ([Fig 5D](#pone.0128240.g005){ref-type="fig"}; lanes 12, 14). Combined with the evidence for K63-linked di-Ub in the proteomics analysis, we propose that the +1.5 band comes from Hsp70-Ub~(2)~ where the first Ub is attached to the N-terminus of the substrate and the second Ub is attached to K63 of the first. Moreover, the other +n.5 bands would represent additional K63-linked extensions of the Ub chain. The aberrant mobility of this species in denaturing PAGE analysis is thus attributed to the forked di-ubiquitin at the N-terminus, which likely causes the protein to move faster through the gel matrix than linear peptides of the same mass. Discussion {#sec011} ========== Our analysis of CHIP ubiquitination products greatly extends knowledge of the activity of this E3 ligase and its specificity towards substrates. Previous studies found only six ubiquitination sites on Hsp70 \[[@pone.0128240.ref009]\], but with the significantly greater coverage in the proteomic analysis we find there are 6 additional sites that are modified. In addition, we report 16 sites are modified on the highly homologous Hsc70. Interestingly, the sites for both Hsc70 and Hsp70 are distributed throughout and on multiple faces of the proteins, and the sites are nearly identical between the two highly conserved proteins. This pattern of Ub modification implies that there are multiple orientations for Ub to transfer to these substrates. Since the ubiquitination machinery has multiple flexible linkers (including the E2-Ub bond, the CHIP dimer, and the Hsx70 C-terminus) it is expected that most of the surface exposed lysines would be available for ubiquitination \[[@pone.0128240.ref018]--[@pone.0128240.ref020]\]. However, only 12 of the 39 observed Hsp70 lysines and 16 of the 45 observed Hsc70 lysines are modified. As we observe good coverage and strong mass spectrometry signals for the unmodified protein, we can infer that most sites are not ubiquitinated at significant levels. Hence, modification does not occur for every accessible site. This implies that the selection of sites of ubiquitination involves multiple factors, not just accessibility to the ubiquitination machinery. Ubiquitin linkage types also reflect the specificity of the ubiquitination machinery. Previous studies have reported that CHIP can produce all possible Ub linkages and that Hsp70 is modified in vitro with K6, K11, K48, and K63-linked chains \[[@pone.0128240.ref008],[@pone.0128240.ref009]\]. Although we also observe diversity in linkage types while noting K48 chains are clearly dominant for Hsp70. This is significantly different from Hsc70, for which the majority of Hsc70 modifications are spread between K6, K11 and K48. In a previous report on Hsc70 ubiquitination in Cos-7 cells \[[@pone.0128240.ref021]\], single lysine mutations K29R and K48R had no apparent effect on ubiquitination, fully consistent with our observations. However, no effect was observed for the K63R mutant in that study, whereas we found a reduction in overall ubiquitination activity. We attribute this difference to the greater sensitivity of our assays to a reduction in the rate of ubiquitination. In contrast, their observation that the Ub double mutant K29R, K63R blocks all but mono-ubiquitination of Hsc70 is not supported by our data. It is possible that this combination of Ub mutations significantly inhibits E2 or E3 function, but importantly, there are substantial differences in the experiment conditions between our studies and the previous report. In this report, by performing experiments under identical conditions, there is very high confidence in the intrinsic differences we find between the predominance of K48 chains from CHIP-mediated ubiquitination of Hsp70 versus the distribution of K6, K11, and K48 chains for Hsc70. The analysis of Ub chain types by proteomics methods versus the effects of ubiquitin mutants on ubiquitination reactions proved highly insightful, largely because these two methods are orthogonal and provide complementary views. Consider the case of Hsp70, for which one might anticipate a considerable change in the extent of ubiquitination of Hsp70 in reactions with the K48R Ub mutant. The fact that the no significant difference in ubiquitination is observed seems to suggest that the proteomic and biochemical approaches do not agree; however, the apparent discrepancy can be explained by the generation of chains on the other lysine sites on ubiquitin when the preferred K48 option is not available. This phenomenon also greatly complicates and effectively precludes detailed interpretation of results obtained in degradation assays with Ub K-R mutants. In our experiments, the extent of degradation of Hsp70 was sufficient to discern differences between the mutants, but this is not the case for Hsc70. The K-R mutation of any single lysine is insufficient to greatly alter degradation of Hsc70 because only a limited portion of the chains that are most effectively recognized and processed by the proteasome are formed with WT Ub and any of the mutants. In contrast, the reduction in degradation of Hsp70 observed for K48R is consistent with the predominance of K48 chains for this substrate and more effective degradation of Hsp70 relative to Hsc70. Mass spectrometry-based proteomics is limited by the fact that the observation of a peptide is highly dependent on multiple factors, including its mass, charge, ionization ability, hydrophilicity, fragmentation characteristics etc. \[[@pone.0128240.ref022],[@pone.0128240.ref023]\]. Moreover, specialized approaches are required to obtain quantitative data and are of limited value for a broad survey such as of the large Hsc70 and Hsp70 proteins. The biochemical assays with Ub mutants are complementary because they provide an overview of the total ubiquitination modification of the substrate; however, these data do not distinguish the different chain types that are present. Hence, it is analysis of the ensemble of data that provides the greatest insight. One concern about Hsx70 ubiquitination by CHIP is that no chains containing \>4 Ub molecules were observed and therefore the ubiquitination reaction may not be processive. This is reflected in the observation of high molecular weight poly-Ub CHIP species, yet no poly-Ub Hsx70 for the same samples (see [S2 Fig](#pone.0128240.s002){ref-type="supplementary-material"}). A similar observation was reported in a previous study of Hsp70 ubiquitination and degradation, which found a correlation between in vitro and in vivo studies \[[@pone.0128240.ref006]\]. Polyubiquitinated Hsx70 has been observed by western blot in other studies but only under conditions with high concentrations of CHIP and where the Ub ladder is not resolved \[[@pone.0128240.ref024]\]. While Ub could interfere with antibody recognition of Hsc70-Ub~(n)~, our controls suggest that the high molecular weight species identified by Coomassie stain in our assays are unaffected by the presence of substrate and thus are primarily ubiquitinated CHIP. The length of Ub chains has important implications for substrate recognition and recruitment to the 26S proteasome \[[@pone.0128240.ref025]\]. For proteins with a well-structured domain, ubiquitination and subsequent recognition by the 19S activator of the proteasome is required. It is commonly believed that a chain of at least four ubiquitin molecules is required for recognition by the 19S activator. Degradation of ubiquitinated substrates with fewer Ub has been reported when the ubiquitination site is adjacent to a significantly unstructured region \[[@pone.0128240.ref026]\]. In fact, mono-ubiquitination may be sufficient for recruitment to the proteasome and several examples have appeared in the literature in the last few years \[[@pone.0128240.ref027]\]. What allows substrates with fewer than four-Ub chains to be degraded in the proteasome? The main benefit of a longer Ub chain is that it allows simultaneous interaction with multiple ubiquitin interaction motifs and consequently will bind tighter to 19S activator proteins. Although a short chain may not bind as strongly, the adjacent unstructured regions of proteins could lower the energy barrier to pull the substrate into the core of the 19S activator. Hsp70 and Hsc70 contain three folded domains with an unstructured C-terminal tail and multiple flexible hinges that facilitate conformational exchange during the ATPase cycle. Additionally, the largest sequence divergence in the two proteins occurs at the C-terminus where Hsp70 has five more residues contributing to the flexible region. Thus, flexibility at the C-terminus of Hsp70 and Hsc70 may not only provide a mechanism for substrate degradation with the addition of only 1--4 Ub molecules upon CHIP ubiquitination, but may also contribute to the differential degradation of these two homologous proteins. The differences we find in CHIP ubiquitination of Hsp70 versus Hsc70 may provide clues for why these two chaperones are degraded at much different rates in the proteasome. Although data that directly show a strong correlation between chain type and degradation by the 26S proteasome is lacking, one can speculate on possible explanations, including one or more of the following factors: (i) subtle differences in which lysine residues on the two proteins are modified; (ii) which Ub-Ub linkage is dominant; (iii) the flexibility of the protein. Also, one cannot discount a key role for one or more lysine residues that are not detected in the proteomic analysis. Most importantly, in extrapolating to the cellular context, additional accessory factors and the role of de-ubiquitinases need to be considered. It is clear that investigations of all of these factors needs to be undertaken to discern how it is possible that the level of these chaperones with such high homology and structural similarity can be appropriately balanced to support protein triage both in cellular homeostasis and during stress response. Conclusion {#sec012} ========== The detailed analysis of ubiquitinated Hsx70 proteins presented here refines knowledge of the ubiquitination activity of CHIP and extends the earlier experiments that first examined CHIP mediated ubiquitination. Our proteomic and in vitro ubiquitination analyses indicate that the CHIP ubiquitination machinery generates a range of ubiquitination modifications of both chaperones. However, within this context specific differences in the ubiquitination of Hsp70 and Hsc70 occur, even though they are highly homologous and structurally similar proteins. Importantly, our results represent one of only very few in-depth comparative studies demonstrating that two highly homologous proteins can be differentially ubiquitinated. In addition, our studies of reactions with Ub mutants confirm an underlying plasticity in the ubiquitination machinery that can compensate for the blockage of certain chain linkages and even alter chain type specificity. Ubiquitinated Hsp70 has modifications spread throughout the protein and primarily K48-linked Ub chains, whereas Hsc70 has a wider range of sites of ubiquitination and of types of Ub chains formed. The larger amount of K48 chains on Hsp70 is intriguing, if one assumes that these lead to higher efficiency of substrate recognition and/or degradation by the 26S proteasome. Additional data on the correlation between the distribution of Ub chain types and degradation would be of great interest to confirm or refute this hypothesis. In summary, our data have provided new insights into the specificity of the CHIP E3 ligase and the sensitivity of ubiquitination machinery to even subtle differences in the sequence and structure of its substrates. Supporting Information {#sec013} ====================== ###### Residue-level details of the proteomics results for the studies with UbcH5a and CHIP with both Hsc70 and Hsp70. (DOCX) ###### Click here for additional data file. ###### High molecular weight ubiquitination products are mostly CHIP-Ub~(n)~. In vitro ubiquitination reactions of 30 minutes without and with substrates were blotted for Hsx70, Ub and CHIP with the respective antibodies as labeled. While high molecular weight species are identified as containing CHIP and Ub, only products containing +1--2 Ub can be observed for the Hsx70 substrates. (TIF) ###### Click here for additional data file. ###### Ub mutants alter the rate of Hsx70 degradation in vitro. In the same manner as [Fig 2](#pone.0128240.g002){ref-type="fig"}, concurrent ubiquitination and degradation reactions with UbcH5a and the Ub type indicated were incubated for 2 hours at 37°C. The amount of unmodified Hsx70 was quantified and the relative amount of protein remaining in each reaction is indicated. (TIF) ###### Click here for additional data file. ###### Ube2W has a unique ubiquitination pattern for substrates of CHIP. Hsp70 ubiquitination reactions were incubated for 30 minutes as above except using Ube2W alone or in combination with Ubc13/Uev1a the E2. Samples for mass spectrometry proteomics were taken from lanes 3, 4, 7, and 13 as noted in the text. (TIF) ###### Click here for additional data file. ###### Residue-level details of the proteomics results for the studies with Ube2W and CHIP with Hsp70. (DOCX) ###### Click here for additional data file. ###### Confirmation of N-terminal ubiquitination of Hsp70 by Ube2W. The annotated MS/MS spectrum for the peptide (gg)-GPGSMAK identifies the signature Ub diGly tag attached directly to the N-terminus of the Hsp70 protein. (TIF) ###### Click here for additional data file. The authors would like to thank David Friedman for advice on initial mass spectrometry proteomics studies. We also acknowledge the support and use of facilities at the Mass Spectrometry Research Center Proteomics Core at Vanderbilt. [^1]: **Competing Interests:**The authors have declared that no competing interests exist. [^2]: Conceived and designed the experiments: SES KLR WJC. Performed the experiments: SES KLR SH SJ. Analyzed the data: SES KLR WJC. Wrote the paper: SES WJC.

On why Collet's doubts regarding the PCA are misplaced. Collet's (1989) doubts regarding the efficacy of principal component analysis (PCA) as a tool in the study of event-related brain potentials (ERPs) are unpersuasive. The substantive point reported by Collet is that data points are autocorrelated over the time period during which a single, PCA-defined, ERP component predominates. Such autocorrelation has long been recognized by investigators in the field, and its confirmation by autoregressive modeling does not provide useful information about issues central to ERP data analysis. Furthermore, because he fails to take a full count of the number of parameters used in his autoregressive model his argument from parsimony is flawed. In any event, Collet's argument misperceives the heuristic mode in which PCA is used in actual studies.

Low-electromagnetic interference (“EMI”) switching systems, or switched circuits, are generally known in the art. These systems employ the use of electronic switches, such as transistors, to rapidly connect and disconnect a load, a power source, a signal, or other electrical circuitry within the system. Often, these systems utilize multiple switches, and often instances exist when one or more switches are to be engaged at a same desired time that one or more other switches are to be disengaged, or visa versa. To cost-effectively control EMI emissions of switching systems, the engaging and disengaging of switches is overlapped using pre-calculated timing in an effort to rid the system of fly-back voltage and shoot-through current without the need for additional external filtering components. The switch overlap can be realized and controlled by dividing each switch into multiple independently-controlled switches in parallel with varying impedances (essentially creating a composite switch). When these parallel switches are operated sequentially, the impedance transition of the composite switch is slowed. Applying this technique to multiple switches and overlapping transitions can effectually eliminate both fly-back voltage and shoot-through current. Additionally, this decrease in high-frequency energy may help result in lower EMI. Although effective for a wide range of output power levels, this technique's performance can be less than optimal when the output power falls outside of the effective power range of the pre-calculated timing values. Particularly, if output power is too low, the overlap time may be too long, resulting in excessive shoot-through current. This excessive shoot-through current may dominate the quiescent current of the system as a whole in low power applications where often it is desired to keep quiescent current to a minimum. Conversely, in high output power applications, the overlap may be too short, resulting in fly-back voltage, and thus defeating the desired low-EMI effect of the circuit.

epsf \#1 23.5cm 1ex -40pt = .5ex i \#1 Ø\#1[O(\#1)]{} \#1[\#1]{} \#1 \#1\#2 \#1 \#1\#2 \#1 \#1\#2 \#1[\_[\#1]{}]{} \#1 \#1 \#1[(\[\#1\])]{} \#1[(\#1)]{} \#1\#2[\_[\#1]{}\^[\#2]{}]{} N[$\frac{1}{N}$ expansion]{} \#1[(\#1)\_]{} \#1\#2[\_[i=1]{}\^N|\_i(\#2)\_[\#1]{}\_i(\#2)]{} \#1[{\#1}\_[PB]{}]{} \#1 [**October ’93**]{}\ [**LOOP EQUATION AND AREA LAW IN TURBULENCE**]{} [**A.A. Migdal**]{} [*Physics Department, Princeton University,\ Jadwin Hall, Princeton, NJ 08544-1000.\ E-mail: migdal@acm.princeton.edu*]{} Introduction ============ Incompressible fluid dynamics underlies the vast majority of natural phenomena. It is described by famous Navier-Stokes equation $$\dot{v}_{\alpha} = \nu \partial_{\beta}^2 v_{\alpha} - v_{\beta} \partial_{\beta} v_{\alpha} - \partial_{\alpha} p \\;\; \partial_{\alpha}v_{\alpha} = 0 \label{eq1}$$ which is nonlinear, and therefore hard to solve. This nonlinearity makes life more interesting, though, as it leads to turbulence. Solving this equation with appropriate initial and boundary conditions we expect to obtain the chaotic behavior of velocity field. The simplest boundary conditions correspond to infinite space with vanishing velocity at infinity. We are looking for the translation invariant probability distribution for velocity field, with infinite range of the wavelengths. In order to compensate for the energy dissipation, we add the usual random force to the equations, with the short wavelength support, corresponding to large scale energy pumping. One may attempt to describe this probability distribution by the Hopf generating functional (the angular bracket denote time averaging, or ensemble averaging over realizations of the random forces) $$Z[J] = \left \langle \exp \left( \int d^3 r J_{\alpha}(r)v_{\alpha}(r)\right) \right \rangle \label{eq2}$$ which is known to satisfy linear functional differential equation $$\dot{Z} = H\left[J,\frac{\delta}{\delta J} \right] Z \label{eq3}$$ similar to the Schrödinger equation for Quantum Field Theory, and equally hard to solve. Nobody managed to go beyond the Taylor expansion in source $ J $ , which corresponds to the obvious chain of equations for the equal time correlation functions of velocity field in various points in space. The same equations could be obtained directly from Navier-Stokes equations, so the Hopf equation looks useless. In this work[^1] we argue, that one could significantly simplify the Hopf functional without loosing information about correlation functions. This simplified functional depends upon the set of 3 periodic functions of one variable $$C : r_{\alpha} = C_{\alpha}(\theta)\\;\; 0< \theta< 2\pi$$ which set describes the closed loop in coordinate space. The correlation functions reduce to certain functional derivatives of our loop functional with respect to $ C(\theta)$ at vanishing loop $ C \rightarrow 0 $. The properties of the loop functional at large loop $ C $ also have physical significance. Like the Wilson loops in Gauge Theory, they describe the statistics of large scale structures of vorticity field, which is analogous to the gauge field strength. In Appendix A we recover the expansion in inverse powers of viscosity by direct iterations of the loop equation. In Appendix B we study the matrix formulation of the equation, which may serve as a basis of the random matrix description of turbulence. In Appendix C we study the reduced dynamics, corresponding to the functional Fourier transform of the loop functional. We argue, that instead of 3D equations one can use the 1D equations for the Fourier loop $P_{\alp}(\theta,t)$. In Appendix D we discuss the relation between the initial data for velocity field and the $P$ field, and we find particular realisation for these initial data in terms of the gaussian random variables. In Appendix E we introduce the generating functional for the scalar products $ P_{\alp}(\theta)P_{\alp}(\theta') $. The advantage of this functional over the original $\Psi[C]$ functional is the smoother continuum limit. In Appendix F we discuss the possible numerical implementations of the reduced loop dynamics. In Appendix G we show uniqueness of the tensor area law within certain class of functionals. In Appendix H we present the modern view at the old problem of the minimal surface. In Appendix I we show that the triple Kolmogorov correlation function corresponds to a vanishing correlation of vorticity with two velocity fields. The Loop Calculus ================= We suggest to use in turbulence the following version of the Hopf functional $$\Psi \left[C \right] = \left \langle \exp \left( \frac{\i }{\nu} \oint dC_{\alpha}(\theta) v_{\alpha}\left(C(\theta)\right) \right) \right \rangle \label{eq4}$$ which we call the loop functional or the loop field. It is implied that all angular variable $\theta$ run from $ 0 $ to $ 2\pi$and that all the functions of this variable are $ 2\pi$ periodic.[^2] The viscosity $ \nu $ was inserted in denominator in exponential, as the only parameter of proper dimension. As we shall see below, it plays the role, similar to the Planck’s constant in Quantum mechanics, the turbulencecorresponding to the WKB limit $ \nu \rightarrow 0 $. [^3] As for the imaginary unit $\i$, there are two reasons to insert it in the exponential. First, it makes the motion compact: the phase factor goes around the unit circle, when the velocity field fluctuates. So, at large times one may expect the ergodicity, with well defined average functional bounded by $1$ by absolute value. Second, with this factor of $\i$, the irreversibility of the problem is manifest. The time reversal corresponds to the complex conjugation of $\Psi$, so that imaginary part of the asymptotic value of $\Psi$ at $t \ra \8$ measures the effects of dissipation. The loop orientation reversal $ C(\theta) \ra C(2\pi - \theta) $ also leads to the complex conjugation, so it is equivalent to the time reversal. This symmetry implies, that any correlator of odd/even number of velocities should be integrated odd/even number of times over the loop, and it must enter with an imaginary/real factor. Later, we shall use this property in the area law. We shall often use the field theory notations for the loop integrals, = ( \_C dr\_v\_ ) \[eq4’\] This loop integral can be reduced to the surface integral of vorticity field \_ = \_v\_-\_v\_ by the Stokes theorem $$\Gamma_C[v] \equiv \oint_C dr_{\alpha}v_{\alpha}= \int_{S} d \sigma_{\mu\nu} \omega_{\mu\nu} \\;\; \partial S = C$$ This is the well-known velocity circulation, which measures the net strength of the vortex lines, passing through the loop $ C $. Would we fix initial loop $ C $ and let it move with the flow, the loop field would be conserved by the Euler equation, so that only the viscosity effects would be responsible for its time evolution. However, this is not what we are trying to do. We take the Euler rather than Lagrange dynamics, so that the loop is fixed in space, and hence $\Psi$ is time dependent already in the Euler equations. The difference between Euler and equations is the time irreversibility, which leads to complex average $\Psi$ in dynamics. It is implied that this field $\Psi\left[C\right]$ is invariant under translations of the loop $ C(\theta) \rightarrow C(\theta)+ const $. The asymptotic behavior at large time with proper random forcing reaches certain fixed point, governed by the translation- and scale invariant equations, which we derive in this paper. The general Hopf functional (\[eq2\]) reduces for the loop field for the following imaginary singular source $$J_{\alpha}(r) = \frac{\i }{ \nu} \oint_C dr'_{\alpha} \delta^3 \left(r'-r \right) \label{eq8}$$ The $\Psi$ functional involves connected correlation functions of the powers of circulation at equal times. = This expansion goes in powers of effective Reynolds number, so it diverges in turbulent region. There, the opposite WKB approximation will be used. Let us come back to the general case of the arbitrary Reynolds number. What could be the use of such restricted Hopf functional? At first glance it seems that we lost most of information, described by the Hopf functional, as the general Hopf source $J$ depends upon 3 variables $ x,y,z $ whereas the loop $C$ depends of only one parameter $ \theta $. Still, this information can be recovered by taking the loops of the singular shape, such as two infinitesimal loops $R_1, R_2 $, connected by a couple of wires The loop field in this case reduces to $$\Psi \left[C \right] \rightarrow \left \langle \exp \left( \frac{\i}{ 2\nu} \Sigma_{\mu\nu}^{R_1}\omega_{\mu\nu}(r_1) +\frac{\i}{ 2\nu} \Sigma_{\mu\nu}^{R_2 } \omega_{\mu\nu}(r_2) \right) \right \rangle$$ where $$\Sigma_{\mu\nu}^R = \oint_R d r_{\nu}r_{\mu}$$ is the tensor area inside the loop $R$. Taking functional derivatives with respect to the shape of $R_1$ and $R_2$ prior to shrinking them to points, we can bring down the product of vorticities at $r_1$ and $r_2$. Namely, the variations yield \_\^R= \_R(d r\_r\_+ r\_d r\_ ) = \_R ( d r\_r\_ -d r\_r\_ ) where integration by parts was used in the second term. One may introduce the area derivative $\fbyf{}{\sigma_{\mu\nu}(r)}$, which brings down the vorticity at the given point $ r $ at the loop. $$-\nu^2 \frac{\delta^2 \Psi \left[C \right]} {\delta \sigma_{\mu\nu}(r_1)\delta \sigma_{\lambda \rho}(r_2)} \ra \left \langle \omega_{\mu\nu}(r_1) \omega_{\lambda \rho}(r_2) \right \rangle$$ The careful definition of these area derivatives are or paramount importance to us. The corresponding loop calculus was developed in[@Mig83] in the context of the gauge theory. Here we rephrase and further refine the definitions and relations established in that work. The basic element of the loop calculus is what we suggest to call the spike derivative, namely the operator which adds the infinitesimal $ \Lambda $ shaped spike to the loop $$D_{\alpha}(\theta,\epsilon) = \int_{\theta}^{\theta+2\epsilon}d \phi \left( 1-\frac{\left|\theta +\epsilon - \phi\right|}{\epsilon } \right) \frac{\delta}{\delta C_{\alpha}(\phi)}$$ The finite spike operator $$\Lambda(r,\theta,\epsilon) = \exp \left( r_{\alpha} D_{\alpha}(\theta,\epsilon) \right)$$ adds the spike of the height $r$. This is the straight line from $ C(\theta) $ to $ C(\theta + \epsilon) + r$, followed by another straight line from $ C(\theta+\epsilon)+r $ to $ C(\theta+2 \epsilon)$, Note, that the loop remains closed, and the slopes remain finite, only the second derivatives diverge. The continuity and closure of the loop eliminates the potential part of velocity; as we shall see below, this is necessary to obtain the loop equation. In the limit $ \epsilon \rightarrow 0 $ these spikes are invisible, at least for the smooth vorticity field, as one can see from the Stokes theorem (the area inside the spike goes to zero as $ \epsilon $). However, taking certain derivatives prior to the limit $ \epsilon \rightarrow 0 $ we can obtain the finite contribution. Let us consider the operator $$\Pi \left(r,r',\theta ,\epsilon \right) = \Lambda \left(r, \theta,\frac{1}{2} \epsilon \right) \Lambda \left(r',\theta,\epsilon \right)$$ By construction it inserts the smaller spike on top of a bigger one, in such a way, that a polygon appears Taking the derivatives with respect to the vertices of this polygon $ r, r' $ , setting $r=r'=0$ and antisymmetrising, we find the tensor operator $$\Omega_{\alpha\beta}(\theta,\epsilon) = -\i \nu D_{\alpha}\left(\theta,\frac{1}{2} \epsilon \right) D_{\beta}\left(\theta,\epsilon \right) - \{\alpha \leftrightarrow\beta\} \label{OM}$$ which brings down the vorticity, when applied to the loop field $$\Omega_{\alpha\beta}(\theta,\epsilon) \Psi \left[C \right] \stackrel{\epsilon \rightarrow 0}{\longrightarrow} \omega_{\alpha\beta}\left(C(\theta)\right)\Psi \left[C \right] \label{eqom}$$ The quick way to check these formulas is to use formal functional derivatives $$\frac{\delta \Psi \left[C \right]}{\delta C_{\alpha}(\theta)} = C'_{\beta}(\theta) \fbyf{\Psi \left[C \right]}{\sigma_{\alp\bet}\left(C(\theta)\right)}$$ Taking one more functional derivative derivative we find the term with vorticity times first derivative of the $ \delta $ function, coming from the variation of $ C'(\theta) $ = ’(-’) + C’\_() C’\_(’) This term is the only one, which survives the limit $ \epsilon \rightarrow 0 $ in our relation (\[eqom\]). So, the area derivative can be defined from the antisymmetric tensor part of the second functional derivative as the coefficient in front of $ \delta'(\theta-\theta') $ . Still, it has all the properties of the first functional derivative, as it can also be defined from the above first variation. The advantage of dealing with spikes is the control over the limit $\eps \ra 0$ , which might be quite singular in applications. So far we managed to insert the vorticity at the loop $ C $ by variations of the loop field. Later we shall need the vorticity off the loop, in arbitrary point in space. This can be achieved by the following combination of the spike operators $$\Lambda \left(r,\theta,\epsilon \right) \Pi \left(r_1,r_2,\theta+\epsilon,\delta \right) \\;\; \delta \ll \epsilon$$ This operator inserts the $ \Pi $ shaped little loop at the top of the bigger spike, in other words, this little loop is translated by a distance $r$ by the big spike. Taking derivatives, we find the operator of finite translation of the vorticity $$\Lambda \left(r,\theta,\epsilon \right) \Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)$$ and the corresponding infinitesimal translation operator $$D_{\mu}(\theta,\epsilon)\Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)$$ which inserts $ \partial_{\mu} \omega_{\alpha \beta} \left( C(\theta) \right) $ when applied to the loop field. Coming back to the correlation function, we are going now to construct the operator, which would insert two vorticities separated by a distance. Let us note that the global $ \Lambda $ spike $$\Lambda \left(r,0,\pi \right) = \exp \left( r_{\alpha}\int_{0}^{2\pi}d \phi \left(1- \frac{ \left|\phi-\pi \right|}{\pi} \right) \frac{\delta}{\delta C_{\alpha}(\phi)}\right)$$ when applied to a shrunk loop $ C(\phi) = 0 $ does nothing but the backtracking from $0$ to $r$ This means that the operator $$\Omega_{\alpha\beta}(0 ,\delta)\Omega_{\lambda \rho}(\pi ,\delta) \Lambda \left(r,0,\pi \right)$$ when applied to the loop field for a shrunk loop yields the vorticity correlation function $$\Omega_{\alpha\beta}(0 ,\delta)\Omega_{\lambda \rho}(\pi ,\delta) \Lambda \left(r,0,\pi \right) \Psi [0] = \left \langle \omega_{\alpha \beta}(0) \omega_{\lambda \rho}(r) \right \rangle$$ The higher correlation functions of vorticities could be constructed in a similar fashion, using the spike operators. As for the velocity, one should solve the Poisson equation $$\partial_{\mu}^2 v_{\alpha}(r) = \partial_{\beta} \omega_{\beta \alpha}(r)$$ with the proper boundary conditions , say, $ v=0 $ at infinity. Formally, $$v_{\alpha}(r) = \frac{1}{\partial_{\mu}^{2}}\partial_{\beta} \omega_{\beta \alpha}(r)$$ This suggests the following formal definition of the velocity operator $$V_{\alpha}(\theta,\epsilon,\delta) = \frac{1}{D_{\mu}^2(\theta,\epsilon)} D_{\beta}(\theta,\epsilon) \Omega_{\beta \alpha}(\theta,\delta)\\;\; \delta \ll \epsilon \label{VOM}$$ $$V_{\alpha}(\theta,\epsilon,\delta)\Psi[C] \stackrel{\delta,\epsilon \rightarrow 0}{\longrightarrow} v_{\alpha} \left(C(\theta) \right) \Psi[C]$$ Another version of this formula is the following integral $$V_{\alpha}(\theta,\epsilon,\delta)= \int d^3 \rho \frac{\rho_{\beta}}{4 \pi |\rho|^3}\Lambda \left(\rho,\theta,\epsilon \right) \Omega_{\alpha\beta}(\theta+ \epsilon ,\delta)$$ where the $ \Lambda $ operator shifts the $ \Omega $ by a distance $ \rho $ off the original loop at the point $ r = C(\theta + \epsilon) $ Loop Equation ============= Let us now derive exact equation for the loop functional. Taking the time derivative of the original definition, and using the Navier-Stokes equation we get in front of exponential $$\oint_C d r_{\alpha} \frac{\i}{ \nu} \left( \nu \partial_{\beta}^2 v_{\alpha} - v_{\beta} \partial_{\beta} v_{\alpha} - \partial_{\alpha} p \right)$$ The term with the pressure gradient yields zero after integration over the closed loop, and the velocity gradients in the first two terms could be expressed in terms of vorticity up to irrelevant gradient terms, so that we find $$\oint_C d r_{\alpha} \frac{\i}{ \nu} \left( \nu \partial_{\beta} \omega_{\beta \alpha} - v_{\beta} \omega_{\beta \alpha} \right) \label{Orig}$$ Replacing the vorticity and velocity by the operators discussed in the previous Section we find the following loop equation (in explicit notations) -i\[C\] = d C\_() ( D\_(,) \_(,) + d\^3 (,,) \_(+ ,)\_(,) ) \[PsiC\] The more compact form of this equation, using the notations of [@Mig83], reads i\[C\] = [H]{}\_C\_C \^2\_[C]{} dr\_ ( i\_ + d\^3 r’ ) \[OLD\] Now we observe that viscosity $ \nu $ appears in front of time and spatial derivatives, like the Planck constant $\hbar$ in Quantum mechanics. Our loop hamiltonian ${\cal H}_C$ is not hermitean, due to dissipation. It contains the second loop derivatives, so it represents a (nonlocal!) kinetic term in loop space. So far, we considered so called decaying turbulence, without external energy source. The energy E = d\^3 r \^2 would eventually all dissipate, so that the fluid would stop. In this case the loop wave function $\Psi$ would asymptotically approach $1$ 1 In order to reach the steady state, we add to the right side of the equation the usual gaussian random forces $f_{\alp}(r,t)$ with the space dependent correlation function = \_(t-t’)F(r-r’) concentrated at at small wavelengths, i.e. slowly varying with $r-r'$. Using the identity = d\^3 r’ F(r-r’) which is valid for arbitrary functional $\Phi$ we find the following imaginary potential term in the loop hamiltonian \_C iU\[C\]= \_[C]{} dr\_\_[C]{} dr’\_ F(r-r’) Note, that orientation reversal together with complex conjugation changes the sign of the loop hamiltonian, as it should. The potential part involves two loop integrations times imaginary constant. The first term in the kinetic part has one loop integration, one loop derivative times imaginary constant. The second kinetic term has one loop integration, two loop derivatives and real constant. The left side of the loop equation has no loop integrations, no loop derivatives, but has a factor of $\i$. The relation between the potential and kinetic parts of the loop hamiltonian depends of viscosity, or, better to say, it depends upon the Reynolds number, which is the ratio of the typical circulation to viscosity. In the viscous limit, when the Reynolds number is small, the loop wave function is close to $1$. The perturbation expansion in $ \inv{\nu}$ goes in powers of the potential, in the same way, as in Quantum mechanics. The second (nonlocal) term in kinetic part of the hamiltonian also serves as a small perturbation (it corresponds to nonlinear term in the equation). The first term of this perturbation expansion is just 1 - |\_C d e\^[ik r]{}|\^2 with $\tilde{F}(k)$ being the Fourier transform of $F(r)$. This term is real, as it corresponds to the two-velocity correlation. The next term comes from the triple correlation of velocity, and this term is purely imaginary, so that the dissipation shows up. This expansion can be derived by direct iterations in the loop space as in [@Mig83], inverting the operator in the local part of the kinetic term in the hamiltonian. This expansion is discussed in Appendix A. The results agree with the straightforward iterations of the equations in powers of the random force, starting from zero velocity. So, we have the familiar situation, like in QCD, where the perturbation theory breaks because of the infrared divergencies. For arbitrarily small force, in a large system, the region of small $k$ would yield large contribution to the terms of the perturbation expansion. Therefore, one should take the opposite WKB limit $\nu \ra 0$. In this limit, the wave function should behave as the usual WKB wave function, i.e. as an exponential The effective loop Action $S[C]$ satisfies the loop space Hamilton-Jacobi equation \[C\] =-iU\[C\] + \_[C]{} dr\_ d\^3 r’ \[SC\] The imaginary part of $S[C]$ comes from imaginary potential $U[C]$, which distinguishes our theory from the reversible Quantum mechanics. The sign of $\Im S$ must be positive definite, since $ |\Psi| <1$. As for the real part of $S[C]$, it changes the sign under the loop orientation reversal $C(\theta) \ra C(2\pi-\theta) $. At finite viscosity there would be an additional term -\_[C]{} dr\_\_ -i\_[C]{} dr\_ d\^3 r’ on the right of . As for the term -\_[C]{} dr\_ i(\_S\[C\]) which formally arises in the loop equation, this term vanishes, since $\partial_{\beta}S[C]=0$. This operator inserts backtracking at some point at the loop without first applying the loop derivative at this point. As it was discussed in the previous Section, such backtracking does not change the loop functional. This issue was discussed at length in , where the Leibnitz rule for the operator $ \dal \fbyf{}{\sigma_{\bet\gam}} $ was established = f’(g\[C\]) In other words, this operator acts as a first order derivative on the loop functional with finite area derivative (so called Stokes type functional). Then, the above term does not appear. The Action functional $ S[C] $ describes the distribution of the large scale vorticity structures, and hence it should not depend of viscosity. In terms of the above connected correlation functions of the circulation this corresponds to the limit, when the effective Reynolds number $\frac{\Gamma_C[v]}{\nu}$ goes to infinity, but the sum of the divergent series tends to the finite limit. According to the standard picture of turbulence, the large scale vorticity structures depend upon the energy pumping, rather than the energy dissipation. It is understood that both time $ t $ and the loop size[^4] $ |C| $ should be greater then the viscous scales $$t \gg t_0 = \nu^{\frac{1}{2}}{\cal E}^{-\frac{1}{2}} \\;\; |C| \gg r_0 = \nu^{\frac{3}{4}} {\cal E}^{-\frac{1}{4}}$$ where $ {\cal E } $ is the energy dissipation rate. It is defined from the energy balance equation 0 = \_t= + which can be transformed to = 3 F(0) The left side represents the energy, dissipated at small scale due to viscosity, and the right side - the energy pumped in from the large scales due to the random forces. Their common value is $\et$. We see, that constant $F(r-r')$, i.e., $\tilde{F}(k)\propto \delta(k)$ is sufficient to provide the necessary energy pumping. However, such forcing does not produce vorticity, which we readily see in our equation. The contribution from this constant part to the potential in our loop equation drops out (this is a closed loop integral of total derivative). This is important, because this term would have the wrong order of magnitude in the turbulent limit - it would grow as the Reynolds number. Dropping this term, we arrive at remarkably simple and universal functional equation \[C\] = \_[C]{} dr\_ d\^3 r’ \[KIN\] The stationary solution of this equation describes the steady distribution of the circulation in the strong turbulence. Note, that the stationary solutions come in pairs $ \pm S$. The sign should be chosen so, that $ \Im S > 0 $, to provide the inequality $ |\Psi| <1$. Scaling law =========== The ‘Hamilton-Jacobi’ equation without the potential term (\[KIN\]) allows the family of the scaling solutions $$S[C] = t^{2 \kappa -1}\phi \left[\frac{C}{t^{\kappa}} \right]$$ with arbitrary index $ \kappa $. The scaling function satisfies the equation $$(2 \kappa -1 ) \phi[C] - \kappa \oint_{C} dr_{\alpha} \frac{\delta \phi[C]}{\delta \sigma_{\beta \alpha}(r)}r_{\beta} = \oint_{C} dr_{\alpha} \int d^3 r'\frac{r'_{\gamma}-r_{\gam}}{4 \pi |r-r'|^3} \frac{\delta \phi[C]}{\delta \sigma_{\beta \alpha}(r)} \frac{\delta \phi[C]}{\delta \sigma_{\beta \gamma}(r')}$$ The left side here was computed, using the chain rule differentiation of functional. Asymptotically, at large time, we expect the fixed point, which is the homogeneous functional $$S_{\infty}[C] = |C|^{2- \frac{1}{\kappa}} f \left[\frac{C}{|C|} \right]$$ zeroing the right side of our ‘kinetic’ functional equation $$0=\oint_{C} dr_{\alpha} \int d^3 r'\frac{r'_{\gamma}-r_{\gam}}{4 \pi |r-r'|^3} \frac{\delta S_{\infty}[C]}{\delta \sigma_{\beta \alpha}(r)} \frac{\delta S_{\infty}[C]}{\delta \sigma_{\beta \gamma}(r')}$$ The Kolmogorov scaling [@Kolm41] would correspond to $$\kappa = \frac{3}{2}$$ in which case one can express the $ S $ functional in terms of $ { \cal E } $ $$S[C] = {\cal E} t^2 \phi \left[\frac{C}{\sqrt{{\cal E}t^3}} \right]$$ One can easily rephrase the Kolmogorov arguments in the loop space. The relation between the energy dissipation rate and the velocity correlator reads $${\cal E } = \left \langle v_{\alpha}(r_0) v_{\beta}(0) \partial_{\beta} v_{\alpha}(0) \right \rangle$$ where the point splitting at the viscous scale $r_0$ is introduced. Such splitting is necessary to avoid the viscosity effects; without the splitting the average would formally reduce to the total derivative and vanish. Instead of the point splitting one may introduce the finite loop of the viscous scale $ |C| \sim r_0 $, and compute this correlator in presence of such loop. This reduces to the WKB estimates $$\omega_{\alpha \beta}(r) \rightarrow \frac{\delta S[C]}{\delta \sigma_{\alpha \beta}(r)} \\;\; v_{\alpha}(r) = \int d^3 r'\frac{r'_{\gamma}-r_{\gam}}{4 \pi |r-r'|^3} \omega_{\alpha \gamma}(r')$$ Using the generic scaling law for $ S $ we find $$\omega \sim r_0^{- \frac{1}{\kappa}}\\;\; v \sim r_0^{1- \frac{1}{ \kappa}}\\;\; {\cal E} \sim r_0^{2 - \frac{3}{\kappa}}$$ We see, that the energy dissipation rate would stay finite in the limit of the vanishing viscous scale only for the Kolmogorov value of the index. This argument looks rather cheap, but I think it is basically correct. The constant value of the energy dissipation rate in the limit of vanishing viscosity arises as the quantum anomaly in the field theory, through the finite limit of the point splitting in the correspondent energy current.[^5] There is another version of this argument, which I like better. The dynamics of Euler fluid in infinite system would not exist, for the non-Kolmogorov scaling. The extra powers of loop size would have to enter with the size $L$ of the whole system, like $\left(\frac{|C|}{L}\right)^{\eps} $. So, in the regime with finite energy pumping rate $\et$ the infinite Euler system can exist only for the Kolmogorov index. This must be the essence of the original Kolmogorov reasoning . The problem is that nobody proved that such limit exists, though. Within the usual framework, based on the velocity correlation functions, one has to prove, that the infrared divergencies, caused by the sweep, all cancel for the observables. Within our framework these problems disappear, as we shall see later. As for the correlation functions in inertial range, unfortunately those cannot be computed in the WKB approximation, since they involve the contour shrinking to a double line, with vanishing area inside. Still, most of the physics can be understood in loop terms, without these correlation functions. The large scale behavior of the loop functional reflects the statistics of the large vorticity structures, encircled by the loop. Loop Equation for the Circulation PDF ===================================== The loop field could serve as the generating function for the PDF $P_C(\Gamma) $ for the circulation. The Fourier integral P\_C() = \_[-]{}\^ can be analyzed in the same way as the loop field before. The only difference is that the factors of $ g $ appear in front of various terms. These factors can be replaced by g i acting on $P_C(\Gamma) $. As a result we find \_C() = -\_[C]{} dr\_ d\^3 r’ + \_[C]{} dr\_\_ - U\[C\] \[PDF\] All the imaginary units disappear, as they should. As for the viscosity and forcing, these terms can be neglected in inertial range in the same way as before. The only new thing is that one has to assume that $ \Gamma \gg \nu $ in inertial range in addition to above assumptions about the size of the loop. In absence of these terms there are no dimensional parameters so that the following scaling laws hold (with the same index $ \kappa $ as before) P\_C() = t\^[2 -1]{} FThe factor $ t^{2 \kappa-1} $ came from the normalization of probability density. Note, that this is more general law than before. Here we do not have to use the WKB approximation for the PDF. In other words, the whole PDF rather than just its decay at large $ \Gamma $ satisfies the scaling law. The steady distribution would have the form of P\_C() where the scaling functional $ \Phi $ satisfies the homogeneous equation \_[C]{} dr\_ d\^3 r’ =0 with the normalization condition 1=\_[-]{}\^ In principle, there could be different scaling functions for positive and negative $ \Gamma $, rather than just absolute value $ |\Gamma| $ prescription. This would correspond to above mentioned violation of the time reversal symmetry. However, as we mentioned above, there is no exact relation which would eliminate the symmetric solution. The Kolmogorov triple correlation function vanishes for vorticities (see Appendix I), so that there is no restriction on the asymmetric part of the circulation PDF. Nevertheless, the Kolmogorov scaling $ \kappa = \frac{3}{2} $ seems to me the most likely possibility, by the reasons discussed in the previous section. The homogeneous loop equation requires some boundary conditions at large loops, to provide a meaningful solution. The asymptotic decrease of PDF P\_C() \~, Q would lead to the same WKB equation as before \_[C]{} dr\_ d\^3 r’ =0 We are studying this equation in the next section. Tensor Area law =============== The Wilson loop in QCD decreases as exponential of the minimal area, encircled by the loop, leading to the quark confinement. What is the similar asymptotic law in turbulence? The physical mechanisms leading to the area law in QCD are absent here. Moreover, there is no guarantee, that $\Psi[C]$ always decreases with the size of the loop. This makes it possible to look for the simple Anzatz, which was not acceptable in QCD, namely S\[C\] = s(\_\^C) where \_\^C= \_C r\_ d r\_ is the tensor area encircled by the loop $C$. The difference between this area and the scalar area is the positivity property. The scalar area vanishes only for the loop which can be contracted to a point by removal of all the backtracking. As for the tensor area, it vanishes, for example, for the $8$ shaped loop, with opposite orientation of petals. Thus, there are some large contours with vanishing tensor area, for which there would be no decrease of the $\Psi$ functional. In QCD the Wilson loops must always decrease at large distances, due to the finite mass gap. Here, the large scale correlations are known to exist, and play the central role in the turbulent flow. So, I see no convincing arguments to reject the tensor area Anzatz. This Anzatz in QCD not only was unphysical, it failed to reproduce the correct short-distance singularities in the loop equation. In turbulence, there are no such singularities. Instead, there are the large-distance singularities, which all should cancel in the loop equation. It turns out, that for this Anzatz the (turbulent limit of the) loop equation is satisfied automatically, without any further restrictions. Let us verify this important property. The first area derivative yields \_\^C(r)= = 2 The factor of $2$ comes from the second term in the variation = \_\_-\_\_ Note, that the right side does not depend on $r$. Moreover, you can shift $r$ aside from the base loop $C$, with proper wires inserted. The area derivative would not change, as the contribution of wires drops. This implies, that the corresponding vorticity $\omega_{\mu\nu}^C(r) $ is space independent, it only depends upon the loop itself. The velocity can be reconstructed from vorticity up to irrelevant constant sterms \^C(r) = \_\^C This can be formally obtained from the above integral representation \^C(r) =d\^3 r’\_\^C \[INTG\] as a residue from the infinite sphere $ R = |r'| \ra \8$. One may insert the regularizing factor $ |r'|^{-\epsilon}$ in $\omega$, compute the convolution integral in Fourier space and check that in the limit $ \epsilon \ra 0^+$ the above linear term arises. So, one can use the above form of the loop equation, with the analytic regularization prescription. Now, the $v\,\omega$ term in the loop equation reads \_C d r\_ \^C(r)\_\^C \_\^C\_\^C\_\^C This tensor trace vanishes, because the first tensor is antisymmetric, and the product of the last two antisymmetric tensors is symmetric with respect to $\alp\gam$. So, the positive and negative terms cancel each other in our loop equation, like the “income” and “outcome” terms in the usual kinetic equation. We see, that there is an equilibrium in our loop space kinetics. From the point of view of the notorious infrared divergencies in turbulence, the above calculation explicitly demonstrates how they cancel. By naive dimensional counting these terms were linearly divergent. The space isotropy lowered this to logarithmic divergency in , which reduced to finite terms at closer inspection. Then, the explicit form of these terms was such, that they all cancelled. This cancellation originates from the angular momentum conservation in fluid mechanics. The large loop $C$ creates the macroscopic eddy with constant vorticity $\omega_{\alp\bet}^C$ and linear velocity $ v^C(r) \propto r$. This is a well known static solution of the equation. The eddy is conserved due to the angular momentum conservation.The only nontrivial thing is the functional dependence of the eddy vorticity upon the shape and size of the loop $C$. This is a function of the tensor area $\Sigma_{\mu\nu}^C$, rather than a general functional of the loop. Combining this Anzatz with the space isotropy and the Kolmogorov scaling law, we arrive at the tensor area law \[AREA\] The universal constant $B$ here must be real, in virtue of the loop orientation symmetry. When the orientation is reversed $C(\theta) \ra C(2\pi-\theta)$, the loop integral changes sign, but its square, which enters here, stays invariant. Therefore, the constant in front must be real. The time reversal tells the same, since [*both*]{} viscosity $\nu$ and the energy dissipation rate $\et$ are time-odd. Therefore, the ratio $\frac{\et}{\nu^3}$ is time-even, hence it must enter $\Psi[C]$ with the real coefficient. Clearly, this coefficient $B$ must be positive, since $ \left|\Psi[C] \right|<1$. Note, however, that we did not prove this law. The absence of decay for large twisted loops with zero tensor area is suspicious. Also, the physics seems to be different from what we expect in turbulence. The uniform vorticity, even a random one, as in this solution, contrasts the observed intermittent distribution. Besides, there clearly must be corrections to the asymptotic law, whereas the tensor area law is [*exact*]{}. This is far too simple. We discussed this unphysical solution mostly as a test of the loop technology. Scalar Area law =============== Let us now study the scalar area law, which is a valid Anzatz for the asymptotic decay of the circulation PDF. The set of equations for the minimal surface is summarized in Appendix A. All we need here is the following representation A d \_(x) d \_(y) where $ L_{\Gamma} =|\Gamma|^{\tq} \et^{-\oq} $. The distance $ (x-y)^2$ is measured in 3-space and integration goes along the minimal surface. It is implied that its size is much larger than $ L_{\Gamma} $. In this limit the integration over, say, $ y $ can be performed along the local tangent plane at $ x $ in small vicinity $ y-x \sim L_{\Gamma} $ , after which the factors of $ L_{\Gamma} $ cancel. We are left then with the ordinary scalar area integral A d\_(x) d \_(y) \^2(x-y) d\^2x In the previous, regularized form the area represents so called Stokes functional, which can be substituted into the loop equation. The area derivative of the area reads = d \_(y) In the local limit this reduces to the tangent tensor d \_(y) \^2(x-y) = t\_(x) It is implied that the point $x$ approaches the contour from inside the surface, so that the tangent tensor is well defined t\_(x) = t\_n\_ - t\_ n\_ Here $ t_{\mu}$ is the local tangent vector of the loop, and $ n_{\nu} $ is the inside normal to the loop along the surface. The second area derivative of the regularized area in this limit is just the exponential = Should we look for the higher terms of the asymptotic expansion at large area we would have to take into account the shape of the minimal surface, but in the thermodynamical limit we could neglect the curvature of the loop and use the planar disk. Let us use the general WKB form of PDF P\_C() = We shall skip the arguments of effective action $ Q $. We find on the left side of the loop equation \_t Q \_ Q - \_t \_ Q On the right side we find the following integrand ((\_[A]{}Q)\^2-\^2\_[A]{}Q) - \_[A]{}Q The last term drops after the $r' $ integration in virtue of symmetry. The leading terms in the WKB approximation on both sides are those with the first derivatives. We find \_t Q \_Q = (\_[A]{}Q)\^2 d\^3 r’ In the last integral we substitute above explicit form of the area derivatives and perform the $ d^3r' $ integration first. In the thermodynamical limit only the small vicinity $ r'-y \sim L_{\Gamma} $ contributes, and we find d\^3 r’ L\_\^2 d \_(y) This integral logarithmically diverges. We compute it with the logarithmic accuracy with the following result d \_(y) The meaning of this integral is the average velocity in the WKB approximation. This velocity is tangent to the loop, up to the next correction terms at large area. Now, the emerging loop integral vanishes due to symmetry t\_ t\_ =0 as the line element $ d \ral $ is directed along the tangent vector $ t_{\alpha} $, and the tangent tensor $ t_{\alpha\beta} $ is antisymmetric. Similar mechanism was used in the tensor area solution, only there the cancellations emerged at the global level, after the closed loop integration. Here the right side of the loop equation vanishes locally, at every point of the loop. Anyway, we see, that the scalar area indeed represents the steady solution of the loop equation in the leading WKB approximation. It might be instructive to compare this solution with another known exact solution of the Euler dynamics, namely the Gibbs solution P\[v\] = For the loop functional it reads = The integral diverges, and it corresponds to the perimeter law r’\_ \^3(r-r’) r\_0\^[-2]{} \_C |dr| where $ r_0 $ is a small distance cutoff. For the PDF it yields P() When the Gibbs solution is substituted into the loop equation, we observe the same thing. Average velocity is tangent to the loop, which leads to vanishing integrand in the loop equation. The difference is that in our case this is true only asymptotically, there are next corrections. The shape of the function $ Q $ is not fixed by this equation in the leading WKB approximation. In a scale invariant theory it is natural to expect the power law Q(, ) (\^[2]{} A\^[1-2]{})\^ \[MuLaw\] There is one more arbitrary index $ \mu$ involved. Even for the Kolmogorov law $ \kappa = \frac{3}{2} $ the $ \Gamma $ dependence remains unknown. Discussion ========== So, we found two asymptotic solutions of the loop equation in the turbulent limit, not counting the Gibbs solution. It remains to be seen, which one (if any) is realized in turbulent flows. The tensor area solution is mathematically cleaner, but its physical meaning contradicts the intermittency paradigm. It corresponds to the uniform vorticity with random magnitude and random direction, rather that the regions of high vorticity interlaced with regions of low vorticity, observed in the turbulent flows. The recent numerical experiments[@Umeki] favor the scalar area rather than the tensor one. Also, the Kolmogorov scaling was observed in these experiments. The Reynolds number was only $ \sim 100 $ which was too small to make any conclusions. We have to wait for the experiments ( real or numerical ) with the Reynolds numbers few orders of magnitude larger. The scalar area is less trivial than the tensor one. The minimal area as a functional of the loop cannot be represented as any explicit contour integral of the Stokes type, therefore it corresponds to infinite number of higher correlation functions present. Moreover, there could be several minimal surfaces for the same loop, as the equations for the minimal surface are nonlinear. Clearly, the one with the least area should be taken. The natural generalization of this solution is the string Anzatz where the sum over all surfaces bounded by the loop is taken P\_C() = \_[S: S=C]{} At large loop the minimal $ Q $ terms will remain. The extremum condition Q = A =0 will be satisfied for the minimal surface. However, the sum over random surfaces is not well defined. The recent studies indicate that the typical closed surfaces degenerate to branched polymers. For the surface bounded by a fixed loop this cannot happen, of course. Still nobody knows how to compute such sums. The loop equation in principle allows to systematically compute the corrections to the area law as the WKB expansion. The WKB solution is incomplete so far. The leading term in the loop equation is annihilated by arbitrary function of the area (scalar or tensor). The similar ambiguity was present in the Gibbs solution, where arbitrary function of the Hamiltonian satisfied the Liouville equation for the velocity PDF. In that case the ambiguity was removed by extra requirement of thermodynamic limit: only the exponential of the hamiltonian would agree with the factorization of the PDF for two remote parts of the system. What could be a similar requirement here? The area of the minimal surface represents the effective volume of the system at large loop. The circulation can be written as a surface integral of vorticity, which makes the circulation an extensive variable at this surface. The average vorticity | = represents an intensive quantity. The thermodynamic limit would then correspond to Q = A q(|) Comparing this with the previous formula for $ Q $ we conclude that $ \mu=1$. In this case Q =A |\^[2]{} In principle, there could be two different laws for positive and negative $ \Gamma $, due to violation of the time reversal invariance Q q\_ A |||\^[2]{} Another line of argument might start with an assumption of decorrelated average vorticity $ \omega_i $ at various parts $ S_i $ of the area $ A_0 $ of the minimal surface. The net circulation, adding up from the large number $ n \sim \frac{A}{A_0} \gg 1 $ of independent random terms $ \omega_i A_0 $ would be a gaussian variable as a consequence of the law of large numbers. We would have then Q \~ = This would agree with the previous estimate at = , A\_0\_i\^2\~A\^[1- ]{} so that Q \^2 A\^[-2]{} The natural assumption here would be that the vorticity variance $ \omega_i$ does not scale with the area $ A $, so that A\_0 \~A\^[1- ]{} The Gaussian behavior (with $ \kappa = \frac{3}{2} $ ) was observed in numerical experiments , but the Reynolds number was too low to make conclusions at this point. There could be a scaling function Q = q(\^[2]{} A\^[-2]{}) which starts linearly and then grows as a power, say, $ q(x) = (1+ a \,x)^{\kappa} $. I suggest that this function should be studied in real and numerical experiments. This would teach us something new about turbulence. Acknowledgments =============== I am grateful to V. Borue, I. Goldhirsh, D. McLaughlin, A. Polyakov and V. Yakhot for stimulating discussions . Loop Expansion ============== Let us outline the method of direct iterations of the loop equation. The full description of the method can be found in [@Mig83]. The basic idea is to use the following representation of the loop functional = 1+\_[n=2]{}\^ {\_C dr\_1\^[\_1]{} …\_C dr\_n\^[\_n]{}}\_ W\^n\_[\_1…\_n]{}(r\_1,…r\_n) \[STOKES\] This representation is valid for every translation invariant functional with finite area derivatives (so called Stokes type functional). The coefficient functions $W$ can be related to these area derivatives. The normalization $\Psi[0]=1 $ for the shrunk loop is implied. In general case the integration points $r_1,\dots r_n$ in are cyclicly ordered around the loop $C$. The coefficient functions can be assumed cyclicly symmetric without loss of generality. However, in case of fluid dynamics, we are dealing with so called abelian Stokes functional. These functionals are characterized by completely symmetric coefficient functions, in which case the ordering of points can be removed, at expense of the extra symmetry factor in denominator = 1+\_[n=2]{}\^ \_C dr\_1\^[\_1]{} …\_C dr\_n\^[\_n]{} W\^n\_[\_1…\_n]{}(r\_1,…r\_n) \[ABEL\] The incompressibility conditions \_[\_k]{}W\^n\_[\_1…\_n]{}(r\_1,…r\_n)=0 \[divv\] does not impose any further restrictions, because of the gauge invariance of the loop functionals. This invariance (nothing to do with the symmetry of dynamical equations!) follows from the fact, that the closed loop integral of any total derivative vanishes. So, the coefficient functions are defined modulo such derivative terms. In effect this means, that one may relax the incompressibility constraints , without changing the loop functional. To avoid confusion, let us note, that the physical incompressibility constrains are not neglected. They are, in fact, present in the loop equation, where we used the integral representation for the velocity in terms of vorticity. Still, the longitudinal parts of $W$ drop in the loop integrals. The loop calculus for the abelian Stokes functional is especially simple. The area derivative corresponds to removal of one loop integration, and differentiation of the corresponding coefficient function = \_[n=1]{}\^ \_C dr\_1\^[\_1]{} …\_C dr\_n\^[\_n]{} \_\^W\^[n+1]{}\_[,\_1…\_n]{}(r,r\_1,…r\_n) \[ABEL’\] where \_\^ \_\_-\_\_ In the nonabelian case, there would also be the contact terms, with $W$ at coinciding points, coming from the cyclic ordering . In abelian case these terms are absent, since $W$ is completely symmetric. As a next step, let us compute the local kinetic term \_C d r\_ \_ Using above formula for the loop derivative, we find = \_[n=1]{}\^ \_C dr\^\_C dr\_1\^[\_1]{} …\_C dr\_n\^[\_n]{} \^2W\^[n+1]{}\_[,\_1…\_n]{}(r,r\_1,…r\_n) \[L\] The net result is the second derivative of $W$ with respect to one variable. Note, that the second term in $\hat{V}_{\mu\nu}^{\alp}$ dropped, as the total derivative in the closed loop integral. As for the nonlocal kinetic term, it involves the second area derivative off the loop, at the point $r'$, integrated over $r'$ with the corresponding Green’s function. Each area derivative involves the same operator $\hat{V}$, acting on the coefficient function. Again, the abelian Stokes functional simplifies the general framework of the loop calculus. The contribution of the wires cancels here, and the ordering does not matter, so that = \_[n=0]{}\^ \_C dr\_1\^[\_1]{} …\_C dr\_n\^[\_n]{} \_\^\_[’’]{}\^[’]{}W\^[n+2]{}\_[,’,\_1…\_n]{}(r,r’,r\_1,…r\_n) Using these relations, we can write the steady state loop equation as follows Here the light dotted lines symbolize the arguments $\alp_k, r_k$ of $W$, the big circle denotes the loop $C$, the tiny circles stand for the loop derivatives, and the pair of lines with the arrow denote the Green’s function. The sum over the tensor indexes and the loop integrations over $r_k$ are implied. The first term is the local kinetic term, the second one is the nonlocal kinetic term, and the right side is the potential term in the loop equation. The heavy dotted line in this term stands for the correlation function $F$ of the random forces. Note that this term is an abelian Stokes functional as well. The iterations go in the potential term, starting with $\Psi[C]=1$. In the next approximation, only the two loop correction $W^2_{\alp_1\alp_2}(r_1,r_2)$ is present. Comparing the terms, we note, that nonlocal kinetic term reduces to the total derivatives due to the space symmetry (in the usual terms it would be $ \VEV{v\omega}$ at coinciding arguments), so we are left with the local one. This yields the equation \^3 \^2 W\^2\_(r-r’) = F(r-r’)\_ modulo derivative terms. The solution is trivial in Fourier space W\^2\_(r-r’) = - \_ Note, that we did not use the transverse tensor P\_(k) = \_- Though such tensor is present in the physical velocity correlation, here we may use $\del_{\alp\bet}$ instead, as the longitudinal terms drop in the loop integral. This is analogous to the Feynman gauge in QED. The correct correlator corresponds to the Landau gauge. The potential term generates the four point correlation $ F \,W^2$. which agrees with the disconnected term in the $W^4$ on the left side W\^4\_[\_1\_2\_3\_4]{}(r\_1,r\_2,r\_3,r\_4) W\^2\_[\_1\_2]{}(r\_1-r\_2)W\^2\_[\_3\_4]{}(r\_3-r\_4) + W\^2\_[\_1\_3]{}(r\_1-r\_3)W\^2\_[\_2\_4]{}(r\_2-r\_4) + W\^2\_[\_1\_4]{}(r\_1-r\_4)W\^2\_[\_2\_3]{}(r\_2-r\_3) In the same order of the loop expansion, the three point function will show up. The corresponding terms in kinetic part must cancel among themselves, as the potential term does not contribute. The local kinetic term yields the loop integrals of $ \d^2 W^3 $, whereas the nonlocal one yields $\hat{V}W^2 \,\hat{V'}W^2$, integrated over $d^3 r'$ with the Greens’s function $ \frac{(r-r')}{4\pi |r-r'|^3}$. The equation has the structure Now it is clear, that the solution of this equation for $W^3$ would be the same three point correlator, which one could obtain (much easier!) by direct iterations of the equation. The purpose of this painful exercise was not to give one more method of developing the expansion in powers of the random force. We rather verified that the loop equations are capable of producing the same results, as the ordinary chain of the equations for the correlation functions. In above arguments, it was important, that the loop functional belonged to the class of the abelian Stokes functionals. Let us check that our tensor area Anzatz \^C\_=\_C d belongs to the same class. Taking the square we find (\^C\_)\^2 = \_C d \_C d r’\_ r’\_ = - \_C d \_C d r’\_ (r-r’)\^2 where the last transformation follows from the fact, that only the cross term in $ (r-r')^2$ yields nonzero after double loop integration. Any expansion in terms of the square of the tensor area reduces, therefore to the superposition of multiple loop integral of the product of $(r_i-r_j)^2 $, which is an example of the abelian Stokes functional. In the limit of large area, this could reduce to the fractional power. An example could be, say = One could explicitly verify all the properties of the abelian Stokes functional. This example is not realistic, though, as it does not have the odd terms of expansion. In the real world such terms are present at the viscous scales. According to our solution, this asymmetry disappears in inertial range of loops (which does not apply to velocity correlators at inertial range, as those correspond to shrunk loops). Matrix Model ============ The Navier-Stokes equation represents a very special case of nonlinear PDE. There is a well known galilean invariance $$v_{\alpha}(r,t) \rightarrow v_{\alpha}(r-u t,t) + u_{\alpha}$$ which relates the magnitude of velocity field with the scales of time and space. [^6] Let us make this relation more explicit. First, let us introduce the vorticity field $$\omega_{\mu\nu} = \partial_{\mu} v_{\nu} -\partial_{\nu} v_{\mu}$$ and rewrite the Navier-Stokes equation as follows $$\dot{v}_{\alpha} = \nu \partial_{\beta} \omega_{\beta \alpha} - v_{\beta}\omega_{\beta \alpha} - \partial_{\alpha} w \\;\; w = p + \frac{v^2}{2}$$ This $ w $ is the well known enthalpy density, to be found from the incompressibility condition $ div v = 0 $, i.e. $$\partial^2 w = \partial_{\alpha}v_{\beta}\omega_{\beta \alpha}$$ As a next step, let us introduce “covariant derivative” operator $$D_{\alpha} = \nu \partial_{\alpha} - \frac{1}{2}v_{\alpha}$$ and observe that $$2 \left[D_{\alpha} D_{\beta} \right] = \nu \omega_{\beta \alpha}$$ $$2 D_{\beta}\left[ D_{\alpha}D_{\beta} \right] + {\it h.c.}= \nu \partial_{\beta} \omega_{\beta \alpha} - v_{\beta}\omega_{\beta \alpha}$$ where $ {\it h.c.}$ stands for hermitean conjugate. These identities allow us to write down the following dynamical equation for the covariant derivative operator $$\dot{D}_{\alpha} = D_{\beta}\left[ D_{\alpha}D_{\beta} \right] - D_{\alpha} W + {\it h.c.}$$ As for the incompressibility condition, it can be written as follows $$\left[D_{\alpha} D^{\dagger}_{\alpha} \right] =0$$ The enthalpy operator $ W = \frac{w}{\nu}$ is to be determined from this condition , or, equivalently $$\left[D_{\alpha} \left[D_{\alpha} W \right] \right] = \left[D_{\alpha}, D_{\beta}\left[ D_{\alpha}D_{\beta} \right]\right]$$ We see, that the viscosity disappeared from these equations. This paradox is resolved by extra degeneracy of this dynamics: the antihermitean part of the $ D $ operator is conserved. Its value at initial time is proportional to viscosity. The operator equations are invariant with respect to the time independent unitary transformations $$D_{\alpha} \rightarrow S^{\dagger}D_{\alpha}S\\;\; S^{\dagger}S = 1$$ and, in addition, to the time dependent unitary transformations with $$S(t) = \exp \left( \frac{1}{2\nu} t u_\beta \left(D_{\beta} - D_{\beta}^{\dagger} \right) \right)$$ corresponding to the galilean transformations. One could view the operator $ D_{\alpha} $ as the matrix $$\left\langle i | D_{\alpha} | j \right \rangle = \int d^3r \psi_i^{\star}(r) \nu \d_{\alp} \psi_j(r)- \frac{1}{2} \psi_i^{\star}(r)v_{\alpha}(r) \psi_j(r)$$ where the functions $ \psi_j(r) $ are the Fourier of Tchebyshev functions depending upon the geometry of the problem. The finite mode approximation would correspond to truncation of this infinite size matrix to finite size $ N $. This is not quite the same as leaving $ N $ terms in the mode expansion of velocity field. The number of independent parameters here is $ O(N^2) $ rather then $ O(N)$. It is not clear whether the unitary symmetry is worth paying such a high price in numerical simulations! The matrix model of Navier-Stokes equation has some theoretical beauty and raises hopes of simple asymptotic probability distribution. The ensemble of random hermitean matrices was recently applied to the problem of Quantum Gravity [@QG], which led to a genuine breakthrough in the field. Unfortunately, the model of several coupled random matrices, which is the case here, is much more complicated then the one matrix model studied in Quantum Gravity. The dynamics of the eigenvalues is coupled to the dynamics of the “angular” variables, i.e. the unitary matrices $ S $ in above relations. We could not directly apply the technique of orthogonal polynomials, which was so successful in the one matrix problem. Another technique, which proved to be successful in QCD and Quantum Gravity is the loop equations. This method, which we are discussing at length in this paper, works in field theory problems with hidden geometric meaning. The turbulence proves to be an ideal case, much simpler then QCD or Quantum Gravity. The Reduced Dynamics ==================== Let us now try to reproduce the dynamics of the loop field by a simpler Anzatz $$\Psi[C] = \left \langle \exp \left( \frac{\i}{\nu}\oint d C_{\alpha}(\theta) P_{\alpha}(\theta) \right) \right \rangle \label{Reduced}$$ The difference with original definition (\[eq4\]) is that our new function $ P_{\alpha}(\theta) $ depends directly on $ \theta $ rather then through the function $ v_{\alpha}(r) $ taken at $ r_{\alpha} = C_{\alpha}(\theta) $. This is the $ d \rightarrow 1 $ dimensional reduction we mentioned before. From the point of view of the loop functional there is no need to deal with field $ v(r) $ , one could take a shortcut. Clearly, the reduced dynamics must be fitted to the Navier-Stokes dynamics of original field. With the loop calculus, developed above, we have all the necessary tools to build this reduced dynamics. Let us assume some unknown dynamics for the $P $ field $$\dot{P}_{\alpha}(\theta) = F_{\alpha}\left(\theta,[P] \right)$$ and compare the time derivatives of original and reduced Anzatz. We find in (\[Reduced\]) instead of (\[Orig\]) $$\frac{\i}{\nu}\oint d C_{\alpha}(\theta) F_{\alpha}\left(\theta,[P]\right)$$ Now we observe, that $P'$ could be replaced by the functional derivative, acting on the exponential in (\[Reduced\]) as follows $$\frac{\delta}{\delta C_{\alpha}(\theta)} \leftrightarrow -\i\nu P'_{\alpha}(\theta)$$ This means, that one could take the operators of the Section 2, expressing velocity and vorticity in terms of the spike operator, and replace the functional derivative as above. This yields the following formula for the spike derivative $$D_{\alpha}(\theta,\epsilon) = -\i\nu \int_{\theta}^{\theta+2 \epsilon} d \phi \left( 1- \frac{\left|\theta + \epsilon - \phi \right|}{\epsilon} \right) P'_{\alpha}(\phi) = -\i\nu \int_{-1}^{1}d \mu \mbox{ sgn}(\mu) P_{\alpha} \left(\theta + \epsilon (1+ \mu) \right) \label{DP}$$ This is the weighted discontinuity of the function $ P(\theta) $, which in the naive limit $ \epsilon \rightarrow 0 $ would become the true discontinuity. However, the function $ P(\theta) $ has in general the stronger singularities, then discontinuity, so that this limit cannot be taken yet. Anyway, we arrive at the dynamical equation for the $P$ field $$\dot{P}_{\alpha} = \nu D_{\beta} \Omega_{\beta \alpha} - V_{\beta} \Omega_{\beta \alpha} \label{Pdot}$$ where the operators $ V , D, \Omega $ of the Section 2 should be regarded as the ordinary numbers, with definition (\[DP\]) of $D$ in terms of $P$. All the functional derivatives are gone! We needed them only to prove equivalence of reduced dynamics to the Navier-Stokes dynamics. The function $ P_{\alpha}(\theta) $ would become complex now, as the right side of the reduced dynamical equation is complex for real $ P_{\alpha}(\theta) $. Let us discuss this puzzling issue in more detail. The origin of imaginary units was the factor of $ \imath $ in exponential of the definition of the loop field. We had to insert this factor to make the loop field decreasing at large loops as a result of oscillations of the phase factors. Later this factor propagated to the definition of the $ P $ field. Our spike derivative $ D $ is purely imaginary for real $ P $, and so is our $ \Omega $ operator. This makes the velocity operator $ V $ real. Therefore the $ D \Omega $ term in the reduced equation (\[Pdot\]) is real for real $ P $ whereas the $ V \Omega $ term is purely imaginary. This does not contradict the moments equations, as we saw before. The terms with even/odd number of velocity fields in the loop functional are real/imaginary, but the moments are real, as they should be. The complex dynamics of $ P $ simply doubles the number of independent variables. There is one serious problem, though. Inverting the spike operator $ D_{\alpha} $ we implicitly assumed, that it was antihermitean, and could be regularized by adding infinitesimal negative constant to $ D_{\alpha}^2 $ in denominator. This, indeed, works perturbatively, in each term of expansion in time, or that in size of the loop, as we checked. However, beyond this expansion there would be a problem of singularities, which arise when $ D_{\alpha}^2(\theta) $ vanishes at some $ \theta $. In general, this would occur for complex $ \theta $, when the imaginary and real part of $ D_{\alpha}^2(\theta) $ simultaneously vanish. One could introduce the complex variable $$e^{\imath \theta}=z\\;\; e^{-\imath \theta}= \frac{1}{z}\\;\; \oint d \theta = \oint \frac{dz}{\imath z}$$ where the contour of $z $ integration encircles the origin around the unit circle. Later, in course of time evolution, these contours must be deformed, to avoid complex roots of $ D_{\alpha}^2(\theta) $. Initial Data ============ Let us study the relation between the initial data for the original and reduced dynamics. Let us assume, that initial field is distributed according to some translation invariant probability distribution, so that initial value of the loop field does not depend on the constant part of $C(\theta)$. One can expand translation invariant loop field in functional Fourier transform $$\Psi[C] = \int DQ\delta^3 \left(\oint d \phi Q(\phi) \right) W[Q] \exp \left( \imath \oint d \theta C_{\alpha}(\theta) Q_{\alpha}(\theta) \right)$$ which can be inverted as follows $$\delta^3 \left( \oint d \phi Q(\phi)\right) W[Q] = \int DC\Psi[C]\exp \left( -\imath \oint d \theta C_{\alpha}(\theta) Q_{\alpha}(\theta) \right)$$ Let us take a closer look at these formal transformations. The functional measure for these integrations is defined according to the scalar product $$(A,B) = \oint \frac{d \theta}{2 \pi} A(\theta) B(\theta)$$ which diagonalizes in the Fourier representation $$A(\theta) = \sum_{-\infty}^{+\infty} A_n e^{\imath n \theta} \\;\;A_{-n} = A_n^{\star}$$ $$(A,B) = \sum_{-\infty}^{+\infty} A_n B_{-n} = A_0 B_0 + \sum_{1}^{\infty} a'_n b'_n + a''_n b''_n\\;\; a'_n = \sqrt{2} \Re A_n,a''_n = \sqrt{2} \Im A_n$$ The corresponding measure is given by an infinite product of the Euclidean measures for the imaginary and real parts of each Fourier component $$DQ = d^3 Q_0 \prod_{1}^{\infty} d^3 q'_n d^3 q''_n$$ The orthogonality of Fourier transformation could now be explicitly checked, as $$\begin{aligned} \lefteqn{\int DC \exp \left( \imath \int d \theta C_{\alpha}(\theta) \left( A_{\alpha}(\theta) - B_{\alpha}(\theta) \right) \right) }\\ \nonumber &=& \int d^3 C_0 \prod_{1}^{\infty} d^3 c'_n d^3 c''_n \exp \left( 2 \pi \imath \left( C_0 \left(A_0-B_0 \right) + \sum_{1}^{\infty} c'_n\left(a'_n - b'_n \right)+ c''_n\left(a''_n - b''_n \right) \right) \right)\\ \nonumber &=&\delta^3\left(A_0-B_0 \right) \prod_{1}^{\infty} \delta^3\left(a'_n - b'_n \right) \delta^3\left(a''_n - b''_n \right)\end{aligned}$$ Let us now check the parametric invariance $$\theta \rightarrow f(\theta)\\;\; f(2\pi) -f(0) = 2\pi \\;\; f'(\theta) >0$$ The functions $ C(\theta) $ and $ P(\theta) $ have zero dimension in a sense, that only their argument transforms $$C(\theta) \rightarrow C \left( f(\theta) \right) \\;\; P(\theta) \rightarrow P\left( f(\theta) \right)$$ The functions $ Q(\theta) $ and $ P'(\theta) $ in above transformation have dimension one $$P'(\theta) \rightarrow f'(\theta) P'\left( f(\theta) \right)\\;\; Q(\theta) \rightarrow f'(\theta) Q \left( f(\theta) \right)$$ so that the constraint on $ Q $ remains invariant $$\oint d \theta Q(\theta) = \oint df(\theta) Q\left( f(\theta) \right)$$ The invariance of the measure is easy to check for infinitesimal reparametrization $$f(\theta) = \theta + \epsilon(\theta)\\;\; \epsilon(2\pi) = \epsilon(0)$$ which changes $C$ and $(C,C)$ as follows $$\delta C(\theta) = \epsilon(\theta) C'(\theta) \\;\; \delta (C,C) = \oint \frac{d \theta}{2\pi} \epsilon(\theta) 2 C_{\alpha}(\theta) C'_{\alpha}(\theta) = -\oint \frac{d \theta}{2\pi}\epsilon'(\theta)C_{\alpha}^2(\theta)$$ The corresponding Jacobian reduces to $$1 - \oint d \theta \epsilon'(\theta) =1$$ in virtue of periodicity. This proves the parametric invariance of the functional Fourier transformations. Using these transformations we could find the probability distribution for the initial data of $$P_{\alpha}(\theta) = - \nu\int_{0}^{\theta} d \phi Q_{\alpha}(\phi)$$ The simplest but still meaningful distribution of initial velocity field is the Gaussian one, with energy concentrated in the macroscopic motions. The corresponding loop field reads $$\Psi_0[C] = \exp \left( -\frac{1}{2} \oint dC_{\alpha}(\theta) \oint dC_{\alpha}(\theta') f\left(C(\theta)-C(\theta')\right) \right)$$ where $ f(r-r') $ is the velocity correlation function $$\left \langle v_{\alpha}(r) v_{\beta}(r') \right \rangle = \left(\delta_{\alpha \beta}- \partial_{\alpha} \partial_{\beta} \partial_{\mu}^{-2} \right) f(r-r')$$ The potential part drops out in the closed loop integral. The correlation function varies at macroscopic scale, which means that we could expand it in Taylor series $$f(r-r') \rightarrow f_0 - f_1 (r-r')^2 + \dots \label{Taylor}$$ The first term $ f_0 $ is proportional to initial energy density, $$\frac{1}{2} \left \langle v_{\alpha}^2 \right \rangle =\frac{d-1}{2} f_0$$ and the second one is proportional to initial energy dissipation rate $${\cal E}_{0} = -\nu \left \langle v_{\alpha} \partial_{\beta}^2 v_{\alpha} \right \rangle = 2 d(d-1) \nu f_1$$ where $ d=3 $ is dimension of space. The constant term in (\[Taylor\]) as well as $ r^2 + r'^2 $ terms drop from the closed loop integral, so we are left with the cross term $ r r' $ $$\Psi_0[C] \rightarrow \exp \left( - f_1 \oint dC_{\alpha}(\theta) \oint dC_{\alpha}(\theta') C_{\beta}(\theta)C_{\beta}(\theta') \right)$$ This is almost Gaussian distribution: it reduces to Gaussian one by extra integration $$\Psi_0[C] \rightarrow {\rm const }\int d^3 \omega \exp \left( -\omega_{\alpha \beta}^2 \right) \exp \left( 2\imath \sqrt{f_1} \omega_{\mu\nu} \oint dC_{\mu}(\theta) C_{\nu}(\theta) \right)$$ The integration here goes over all $ \frac{d(d-1)}{2} =3 $ independent $ \alpha < \beta $ components of the antisymmetric tensor $ \omega_{\alpha \beta} $. Note, that this is ordinary integration, not the functional one. The physical meaning of this $ \omega $ is the random constant vorticity at initial moment. At fixed $ \omega $ the Gaussian functional integration over $ C $ $$\int DC \exp \left( \imath \oint d \theta \left(\frac{1}{\nu} C_{\beta}(\theta) P'_{\beta}(\theta) +2 \sqrt{f_1} \omega_{\alpha \beta} C'_{\alpha}(\theta)C_{\beta}(\theta) \right) \right)$$ can be performed explicitly, it reduces to solution of the saddle point equation $$P'_{\beta}(\theta) = 4\nu\sqrt{f_1}\omega_{\beta \alpha} C'_{\alpha}(\theta)$$ which is trivial for constant $ \omega $ $$C_{\alpha}(\theta) = \frac{1}{4\nu\sqrt{f_1}} \omega^{-1}_{\alpha \beta} P_{\beta}(\theta)$$ The inverse matrix is not\* unique in odd dimensions, since $ \mbox{Det } \omega_{\alpha\beta} = 0 $. However, the resulting pdf for $ P $ is unique. This is the Gaussian probability distribution with the correlator $$\left \langle P_{\alpha}(\theta) P_{\beta}(\theta') \right \rangle = 2\imath \nu\sqrt{f_1} \omega_{\alpha \beta} {\rm sign}(\theta'-\theta) \label{Corr}$$ Note, that antisymmetry of $ \omega $ compensates that of the sign function, so that this correlation function is symmetric, as it should be. However, it is antihermitean, which corresponds to purely imaginary eigenvalues. The corresponding realization of the $ P$ functions is complex! Let us study this phenomenon for the Fourier components. Differentiating the last equation with respect to $ \theta $ and Fourier transforming we find $$\left \langle P_{\alpha,n} P_{\beta,m} \right \rangle = \frac{4\nu}{m} \delta_{-n m} \sqrt{f_1}\omega_{\alpha \beta}$$ This cannot be realized at complex conjugate Fourier components $ P_{\alpha,-n} = P_{\alpha,n}^{\star} $ but we could take $\bar{P}_{\alpha,n} \equiv P_{\alpha,-n} $ and $ P_{\alpha,n} $ as real random variables, with correlation function $$\left \langle \bar{P}_{\alpha,n}P_{\beta,m} \right \rangle = \frac{4\nu}{m}\delta_{n m}\sqrt{f_1} \omega_{\alpha \beta} \\;\; n>0$$ The trivial realization is $$\bar{P}_{\alpha,n} =\frac{4\nu}{n} \sqrt{f_1}\omega_{\alpha \beta} P_{\beta,n}$$ with $P_{\beta,n} $ being Gaussian random numbers with unit dispersion. As for the constant part $ P_{\alpha,0} $ of $ P_{\alpha}(\theta) $ , it is not defined, but it drops from equations in virtue of translational invariance. W-functional ============ The difficulties of turbulence are hidden in the loop equation, but they show up, if you try to solve it numerically. The main problem is that one cannot get rid of the cutoffs $ \epsilon, \delta \rightarrow 0 $ in the definitions of the spike derivatives. These cutoffs are designed to pick up the singular contributions in the angular integrals, but with finite number of modes, such as Fourier harmonics there would be no singularities. We did not find any way to truncate degrees of freedom in the $ P $ equation, without violating the parametric invariance. It very well may be, that this invariance would be restored in the limit of large number of modes, but it looks that there are too much ambiguity in the finite mode approximation. After some attempts, we found the simpler version of the loop functional, which can be studied analytically in the turbulent region. This is the generating functional for the scalar products $ P_{\alpha}(\theta_1)P_{\alpha}(\theta_2) $ $$W[S] = \left \langle \exp \left( - \oint d \theta_1 \oint d \theta_2 S(\theta_1,\theta_2)P_{\alpha}(\theta_1)P_{\alpha}(\theta_2) \right) \right \rangle \label{W1}$$ where, as before, the averaging goes over initial data for the $P $ field. The time derivative of this W-functional $$\dot{W} = -2 \left \langle \oint d \theta_1 \oint d \theta_2 S(\theta_1,\theta_2)P_{\alpha}(\theta_1)\dot{P}_{\alpha}(\theta_2) \exp \left( - \oint d \theta_1 \oint d \theta_2 S(\theta_1,\theta_2)P_{\alpha}(\theta_1)P_{\alpha}(\theta_2) \right) \right \rangle \label{W2}$$ can be expressed in terms of functional derivatives of $W$ by replacing $$P_{\alpha}(\phi_1)P_{\alpha}(\phi_2) \rightarrow \frac{\delta}{\delta S(\phi_1,\phi_2)} \label{W3}$$ for every scalar product of $P$ fields, which arise after expansion of the spike derivatives (\[DP\]), (\[OM\]), (\[VOM\]) in the scalar product $$P_{\alpha}(\theta_1)\dot{P}_{\alpha}(\theta_2) = \nu P_{\alpha}(\theta_1) D_{\beta}(\theta_2) \Omega_{\beta \alpha}(\theta_2) - P_{\alpha}(\theta_1) V_{\beta}(\theta_2)\Omega_{\beta \alpha}(\theta_2) \label{PP}$$ This equation has the structure $$\dot{W} = \oint d^2 \theta S(\theta_1,\theta_2) \left( A_2 \left[\frac{\delta}{\delta S} \right]W + A_3 \left[\frac{\delta}{\delta S} \right]D^{-2}(\theta,\epsilon)W \right) \label{W4}$$ where $A_k \left[X \right] $ stands for the $k-$ degree homogenous functional of the function $ X(\theta_1,\theta_2) $. The operator $ D^{-2} $ is also the homogeneous functional of the negative degree $ k = -1 $. It can be written as follows $$D^{-2}(\theta,\epsilon)W[S] =\int_{0}^{\infty}d \tau W[S + \tau U]$$ with $$U(\theta_1,\theta_2) = \epsilon^{-2} \mbox{ sgn}(\theta+ \epsilon-\theta_1) \mbox{ sgn}(\theta+ \epsilon-\theta_2)$$ Possible Numerical Implementation ================================= The above general scheme is fairly abstract and complicated. Could it lead to any practical computation method? This would depend upon the success of the discrete approximations of the singular equations of reduced dynamics. The most obvious approximation would be the truncation of Fourier expansion at some large number $ N $. With Fourier components decreasing only as powers of $ n $ this approximation is doubtful. In addition, such truncation violates the parametric invariance which looks dangerous. It seems safer to approximate $ P(\theta) $ by a sum of step functions, so that it is piecewise constant. The parametric transformations vary the lengths of intervals of constant $ P(\theta) $, but leave invariant these constant values. The corresponding representation reads $$P_{\alpha}(\theta) = \sum_{l=0}^{N} \left(p_{\alpha}(l+1)-p_{\alpha}(l) \right) \Theta \left(\theta-\theta_l \right)\\;\; p(N+1) = p(1),\; p(0) = 0 \label{Thetas}$$ It is implied that $ \theta_0 =0 < \theta_1 < \theta_2 \dots < \theta_N < 2\pi $. By construction, the function $ P(\theta) $ takes value $p(l)$ at the interval $ \theta_{l-1} < \theta < \theta_{l} $. We could take $ \dot{P}(\theta) $ at the middle of this interval as approximation to $ \dot{p}(l) $. $$\dot{p}(l) \approx \dot{P}(\bar{\theta}_l)\\;\; \bar{\theta}_l = \frac{1}{2}\left(\theta_{l-1} + \theta_l \right)$$ As for the time evolution of angles $ \theta_l $ , one could differentiate (\[Thetas\]) in time and find $$\dot{P}_{\alpha}(\theta) = \sum_{l=0}^{N} \left(\dot{p}_{\alpha}(l+1)-\dot{p}_{\alpha}(l) \right) \Theta \left(\theta-\theta_l \right) - \sum_{l=0}^{N} \left(p_{\alpha}(l+1)-p_{\alpha}(l) \right) \delta(\theta-\theta_l)\dot{\theta_l}$$ from which one could derive the following approximation $$\dot{\theta_l} \approx \frac{\left(p_{\alpha}(l)-p_{\alpha}(l+1) \right)}{ \left(p_{\mu}(l+1)-p_{\mu}(l)\right)^2} \int_{\bar{\theta}_l}^{\bar{\theta}_{l+1}} d \theta \dot{P}_{\alpha}(\theta)$$ The extra advantage of this approximation is its simplicity. All the integrals involved in the definition of the spike derivative (\[DP\]) are trivial for the stepwise constant $ P(\theta) $. So, this approximation can be in principle implemented at the computer. This formidable task exceeds the scope of the present work, which we view as purely theoretical. Uniqueness of the tensor area law ================================= Let us address the issue of the uniqueness of the tensor area solution. Let us take the following Anzatz S\[C\] = f( \_C d \_C d r’\_ W(r-r’) ) When substituted into the static loop equation (with the area derivatives computed in Appendix A), it yields the following equation for the correlation function $W(r)$ 0=\_C d \_C d r’\_ \_C d r”\_ U\_(r,r’,r”) U\_(r,r’,r”)= W(r-r’)\^\_W(r’-r”) + \^\_ = \_ \_ -\_ \_ \[U\] The derivative $f'$ of the unknown function drops from the static equation. This equation should hold for arbitrary loop $C$. Using the Taylor expansion for the Stokes type functional , we can argue, that the coefficient function $U$ must vanish up to the total derivatives. An equivalent statement is that the third area derivative of this functional must vanish. Using the loop calculus (see Appendix A) we find the following equation 0=\^\_ \^[’]{}\_[’’]{} \^[”]{}\_[””]{} U\_[’”]{}(r,r’,r”) which should hold for arbitrary $r,r',r''$. This leads to the overcomplete system of equations for $W(r)$ in general case. However, for the special case $ W(r) = r^2$ which corresponds to the square of the tensor area \_\^2 = - \_C d \_C d r’\_(r-r’)\^2 the system is satisfied as a consequence of certain symmetry. In this case we find in the loop equation 2 \_C d \_C d r’\_(r-r’)\^2 \_C d r”\_ ( r’\_ - r”\_) \^C\_\_C d \_C d r’\_(r-r’)\^2 The last integral is symmetric with respect to permutations of $ \alp , \bet$, whereas the first factor $ \Sigma^C_{\alp\bet}$ is antisymmetric, hence the sum over $\alp\bet$ yields zero, as we already saw above. It was assumed in above arguments, that the loop $C$ consist of only one connected part. Let us now consider the more general situation, with arbitrary number $n$ of loops $C_1,\dots C_n$. The corresponding Anzatz would be S\_n= s\_n(\^1,…\^n) where $\Sigma^i$ are tensor areas. This function should obey the same WKB loop equations in each variable. Introducing the loop vorticities \^k\_ = 2 which are constant on each loop, we have to solve the following problem. What are the values of $ \omega^k_{\mu\nu}$ such that the single velocity field $\val(r) $ could produce them? We do not see any other solutions, but the trivial one, with all equal $ \omega^k_{\mu\nu} $ and linear velocity, as before. This would correspond to s\_n(\^1,…\^n)= s\_1()\ ; \_=\_[k=1]{}\^n \^k\_= \_[C\_k]{} r\_ d r\_ The loop equation would be satisfied like before, with $ C = \uplus C_k $. This corresponds to the additivity of loops S\_n= S\_1Note, that such additivity is the opposite to the statistical independence, which would imply that S\_n= S\_1The additivity could also be understood as a statement, that any set of $n$ loops is equivalent to a single loop for the abelian Stokes functional. Just connect these loops by wires, and note that the contribution of wires cancels. So, if the area law holds for [*arbitrary*]{} single loop, than it must be additive. This assumption may not be true, though, as it often happens in the WKB approximation. There is no single asymptotic formula, but rather collections of different WKB regions, with quantum regions in between. In our case, this corresponds to the following situation. Take the large circular loop, for which the WKB approximation holds, and try to split it into two large circles. You will have to twist the loop like the infinity symbol $ \8$, in which case it intersects itself. At this point, the WKB approximation might break, as the short distance velocity correlation might be important near the self-intersection point. This may explain the paradox of the vanishing tensor area for the $\8$ shaped loop. From the point of view of our area law such loop is not large at all. Minimal surfaces ================ Let us present here the modern view at the classical theory of the minimal surfaces. The minimal surface can be described by parametric equation S: = X\_(\_1,\_2) The function $ X_{\alpha}(\xi) $ should provide the minimum to the area functional A\[X\] = \_S = d\^2 where G\_[ab]{} = \_a X\_\_b X\_, is the induced metric. For the general studies it is sometimes convenient to introduce the unit tangent tensor as an independent field and minimize A= d\^2 (e\_[ab]{} \_a X\_ \_b X\_ t\_ + (1- t\_\^2 ) ) From the classical equations we will find then t\_ = \_a X\_ \_b X\_; t\_\^2 = 1, which shows equivalence to the old definition. For the actual computation of the minimal area it is convenient to introduce the auxiliary internal metric $ g_{ab} $ A= \_S d\^2 . \[gG\] The straightforward minimization with respect to $ g_{ab} $ yields g\_[ab]{} = 2 G\_[ab]{}, which has the family of solutions g\_[ab]{} = G\_[ab]{}. The local scale factor $ \lambda $ drops from the area functional, and we recover original definition. So, we could first minimize the quadratic functional with respect to $ X(\xi) $ (the linear problem), and then minimize with respect to $ g_{ab} $ (the nonlinear problem). The crucial observation is the possibility to choose conformal coordinates, with the diagonal metric tensor g\_[ab]{} = \_[ab]{} , g\^[-1]{}\_[ab]{} = , = ; after which the local scale factor $\rho$ drops from the integral A\[X,\] = \_S d\^2 \_a X\_ \_a X\_. However, the $ \rho $ field is implicitly present in the problem, through the boundary conditions. Namely, one has to allow an arbitrary parametrization of the boundary curve $ C $. We shall use the upper half plane of $ \xi$ for our surface, so the boundary curve corresponds to the real axis $ \xi_2 = 0 $. The boundary condition will be X\_(\_1,+0) = C(f(\_1)), \[Bcon\] where the unknown function $ f(t) $ is related to the boundary value of $ \rho$ by the boundary condition for the metric g\_[11]{} = = G\_[11]{} = (\_1 X\_)\^2 = C\_’\^2 f’\^2 As it follows from the initial formulation of the problem, one should now solve the linear problem for the $ X $ field, compute the area and minimize it as a functional of $ f(.) $. As we shall see below, the minimization condition coincides with the diagonality of the metric at the boundary \_[\_2=+0]{} = 0 \[Diag\] The linear problem is nothing but the Laplace equation $ \d^2 X = 0 $ in the upper half plane with the Dirichlet boundary condition . The solution is well known X\_() = \_[-]{}\^[+]{} The area functional can be reduced to the boundary terms in virtue of the Laplace equation A\[f\] = d\^2 \_a (X\_ \_a X\_) = -\_[-]{}\^[+]{} d \_1 \_[\_2 = +0]{} Substituting here the solution for $ X $ we find A\[f\] = - \_[-]{}\^[+]{}d t \_[-]{}\^[+]{} d t’ This can be rewritten in a nonsingular form A\[f\] = \_[-]{}\^[+]{}d t \_[-]{}\^[+]{} d t’ which is manifestly positive. Another nice form can be obtained by integration by parts A\[f\] = \_[-]{}\^[+]{}d t f’(t) \_[-]{}\^[+]{} d t’ f’(t’) C’\_(f(t)) C’\_(f(t’)) | t-t’| This form allows one to switch to the inverse function $ \tau(f) $ which is more convenient for optimization A\[\] = \_[-]{}\^[+]{}d f \_[-]{}\^[+]{} d f’ C’\_(f) C’\_(f’) |(f)-(f’)| In the above formulas it was implied that $ C(\8) = 0 $. One could switch to more traditional circular parametrization by mapping the upper half plane inside the unit circle \_1 + i\_2 = i ; = r e\^[i]{} ; r 1. The real axis is mapped at the unit circle. Changing variables in above integral we find X\_(r,) = \_[-]{}\^ C\_(()) (- ) Here () = f(). The last term represents an irrelevant translation of the surface, so it can be dropped. The resulting formula for the area reads A\[\] = \_[-]{}\^ d \_[-]{}\^ d ’ or, after integration by parts and inverting parametrization A\[\] = \_[-]{}\^ d \_[-]{}\^ d ’ C’\_()C’\_(’) | | Let us now minimize the area as a functional of the boundary parametrization $ f(t) $ (we shall stick to the upper half plane). The straightforward variation yields 0 = \_[-]{}\^[+]{} d t’ \[NL\] which duplicates the above diagonality condition . Note that in virtue of this condition the normal vector $ n_{\mu}(x) $ is directed towards $ \d_2 X_{\mu} $ at the boundary. Explicit formula reads n\_(C(f(t))) \_[-]{}\^[+]{} d t’ Let us have a closer look at the remaining nonlinear integral equation . In terms of inverse parametrization it reads 0 = \_[-]{}\^[+]{} d f Introduce the vector set of analytic functions F\_(z) = \_[-]{}\^[+]{} which decrease as $ z^{-2} $ at infinity. The discontinuity at the real axis F\_(+ i0) = C\_’(f) f’() Which provides the implicit equation for the parametrization $ f(\tau) $ d F\_(+i0) = C\_(f) We see, that the imaginary part points in the tangent direction at the boundary. As for the boundary value of the real part of $ F_{\mu}(\tau) $ it points in the normal direction along the surface F\_ n\_ Inside the surface there is no direct relation between the derivatives of $ X_{\mu}(\xi) $ and $ F_{\mu}(\xi) $. The integral equation reduces to the trivial boundary condition F\_\^2(t+i0) = F\_\^2(t-i0) In other words, there should be no discontinuity of $ F_{\mu}^2 $ at the real axis. The solution compatible with analyticity in the upper half plane and $ z^{-2} $ decrease at infinity is F\_\^2(z) = (1+ )\^4P(); = where $ P(\omega) $ defined by a series, convergent at $ |\omega| \le 1 $. In particular this could be a polynomial. The coefficients of this series should be found from an algebraic minimization problem, which cannot be pursued forward in general case. The flat loops are trivial though. In this case the problem reduces to the conformal transformation mapping the loop onto the unit circle. For the unit circle we have simply C\_1 + iC\_2 = ; F\_1 = iF\_2 = - ; P = 0. Small perturbations around the circle or any other flat loop can be treated in a systematic way, by a perturbation theory. Kolmogorov triple correlation and time reversal =============================================== Are there any restrictions on the circulation PDF from the known asymmetry of velocity correlations, in particular, the Kolmogorov triple correlation? The answer is that the Kolmogorov correlation does not imply the asymmetry of [*vorticity*]{} correlations. Taking the tensor version of the $\frac{4}{5}$ law in arbitrary dimension $d$ v\_(0) v\_(0) v\_(r)= ( \_ r\_ + \_ r\_ - \_ r\_ ) \[KOLM\] and differentiating, we find that = 0 So, the odd vorticity correlations could, in fact, be absent, in spite of the asymmetry of the velocity distribution. [99]{} A.N.  Kolmogorov. [*The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers*]{}. C. R. Acad. Sci. URSS, 30(4):301–305, 1941. A. Migdal, [*Loop equations and $\inv{N}$ expansion*]{}, Physics Reports , 201 , 102,(1983). D. Gross, A. Migdal, Physical Review Letters, 717, [**64**]{}, (1990), and Nuclear Physics B340, 333,(1990), M. Douglas and S. Shenker, Nuclear Physics B335, 635, (1990), E. Brezin, V. Kazakov, Physical Letters, 144, [**236B**]{}, (1990). A.A. Migdal, [*Loop equation in turbulence*]{}, PUPT-1383, March ’93, hep-th/9303130. A.A. Migdal, [ *Turbulence as statistics of vortex cells*]{}, PUPT-1409, hep-th/9306152 Makoto Umeki, Tokyo University Preprint , July 93, hep-th/9307144 [^1]: see also where this approach was initiated and where its relation with the generalized Hamiltonian dynamics and the Gibbs-Boltzmann statistics was established [^2]: This parametrization of the loop is a matter of convention, as the loop functional is parametric invariant. [^3]: One could also insert any numerical parameter in exponential, but this factor could be eliminated by space- and/or time rescaling. [^4]: As a measure of the loop size one may take the square root of the minimal area inside the loop. [^5]: I am grateful to A. Polyakov and E. Siggia for inspiring comments on this subject. [^6]: At the same time it tells us that the constant part of velocity if frame dependent, so that it better be eliminated, if we would like to have a smooth limit at large times. Most of notorious large scale divergencies in turbulence are due to this unphysical constant part.

Q: Python: Looping through a list of dictionaries and extracting values to a new dictionary I need to loop on a list of dictionaries and check if a value exist. If it exists then I take another value from this same dictionary and store it on a new dictionary inside another list. What I have is this class_copy=[] for root, dirs, files in os.walk(files_path+"/TD"): for file in files: file_name=os.path.splitext(file)[0] for d in data_list: if d['id'] == file_name: cc['class']=d['fic'] class_copy.append(cc) break So I loop through some files. data_list is a list of dictionaries. These dictionaries each have an 'id' which matches a file name so when the dictionary 'd' with the value 'id' is found I take the value of 'fic' on dictionary 'd' and make a new dictionary with the key 'class' to store the value of 'fic'. Then I store this dictionary on a new list of dictionaries called class_copy. The problem is that after looping through all, all the dictionaries in class_copy are the same. I guess that by looping and changing the instance of d the values on class_copy are also changing but how could I retain the values? Is there a better way to do this? A: You are not actually creating a new dictionary, you are updating an existing one: cc['class']=d['fic'] simply updates the value associated with the key 'class'. Change your code to this: cc = {'class': d['fic']} which will create a new instance of a dictionary. The reason that all entries in your list end up being the same is because each item in the list is the same dictionary. By creating a new dictionary as shown you will have independent instances in the list as you were expecting.

Q: Can I use a ultrasound sensor to measure water level? If I use an ultrasonic sensor will it detect the water level? I was thinking about a product to read water level on water boxes (common in Brazil). I researched about instrumentation for this measure, and I think that an ultrasonic sensor is the best option. Will the water correctly reflect ultrasound and not change the normal measurements against a solid obstacle? A: Technology used in depth finder, in marine application is mostly like is the best to measure the water from the top. To measure water level from the bottom of tank one could use a piezo electric ceramic transducer combined with an Analog Front End (AFE) and a micro-controller to measure water level. The diagram below best explains the configuration. You can use a piezo electric ceramic transducer from Steminc, TDC1000 AFE from Texas Instrument and a MSP430 micro-controller also from Texas Instrument. There might be other configurations, but currently I am only aware of this configuration. Piezo Electric Transducer Analog Front End (AFE) Click on image for a larger version of the image. Limitation: The thickness and material of the tank might be an issue. For the most part above configuration work with plastic. References: Steminc - piezo electric transducer Piezoelectric matching/backing layer materials Piezoelectric Modes of Vibration What is Radial and Thickness Mode Vibration in a Piezo Electric Ceramic Disc Transducer? TDC1000 Ultrasonic Sensing Analog Front End (AFE) for Level Sensing How to measure output of the ultrasonic transducer? A: Yes To answer your specific question, yes, you can use an ultrasonic sensor to measure fluid level. Mahendra's answer describes that. You mentioned that you feel that an ultrasonic sensor is the best solution, but I wanted to add make sure that you were aware of some of the other methods that are used to measure fluid levels. Measuring Techniques This list of fluid level measuring techniques comes from an appropriately named website, "A Dozen Ways to Measure Fluid Level and How They Work". Sight Glass Floats Hydrostatic Devices Bubblers Load Cells Strain Gauges Magnetic Level Gauges Capacitance Transmitters Magnetostrictive Level Transmitters Ultrasonic Level Transmitters Laser Level Transmitters Radar Level Transmitters There is no need to repeat the details of how each of these methods work, but this is a comprehensive list of technologies that could be considered.

Q: Input redirection in C programming? I am trying to redirect a data file called data1 into my program, but I keep on getting a segmentation fault. When I try ./w data1 then it reads it correctly but when I do ./w < data1 then the error pops up. I have to use the second way for my assignment. Here is what my code looks like: int main(int argc, char *argv[]) { FILE *Q; Q = fopen(argv[1],"r"); } A: argv[1] points to the first parameter of your binary (without it's name), so in case of ./w < data1 it is missing. You're trying to access the "illegal" memory, so you get a segfault. The mark < is a bash feature, it is not passed to C. If you want to use such a redirection, just read from standard input and don't care about the file/argv. It means "take file data1 and pass it to descriptor 0, that is standard input". You can use scanf or read(0, ...) to use the file's content. A: The reason your first command works is because you are providing an argument to your program (in your case, the filename) and your program uses that argument to open the file. When you are using a redirect, you are redirecting stdin and stdout (which are accessed with functions like gets and printf) to the file, so you don't have to open any files in the program at all.

Q: Get UserId of shared calendar's owner using MS Graph API I want to get the User's profile that shared a Calendar with me but this call needs the id or userPrincipalName: GET /users/{id | userPrincipalName} The shared calendar only returns: { "id": "**********************************************=", "name": "Lala Lalala", "color": "auto", "changeKey": "Epg+nQ9k3kuTN16cfoLtwAAAsZgDvA==", "canShare": false, "canViewPrivateItems": false, "canEdit": true, "owner": { "name": "Lala Lalala", "address": "Lalala@outlook.com" } } So how can I get the id or userPrincipalName of the shared calendar's owner? A: For Work/School Accounts (Azure AD tenants), the userPrincipalName is the owner's address (i.e. alias@aad.domain.com): "owner": { "name": "Lala Lalala", "address": "Lalala@outlook.com" } Assuming they're in the same tenant as your, you can retrieve their profile using GET https://graph.microsoft.com/v1.0/users/{owner.address}. Important: This does not, however, apply to Personal Accounts (MSA/Outlook.com). I only mention this because your example used Lalala@outlook.com as the address. Since Outlook.com is effectively a "single user" tenant, the only user you can retrieve is yourself (/me). Just as you cannot access a user's data from another company's AAD, you cannot retrieve another Outlook.com user's profile. If you consider the pricacey implications of my access your personal contact information, it makes sense why this rule is in place.

1. Background {#sec1} ============= Insomnia is common amongst the elderly, and in some countries, the reported prevalence rate is over 60% \[[@B1]\]. Elderly people have difficulty falling asleep and maintaining sleep due to frequent awakenings \[[@B1]\]. Sleep loss in the aging population is associated with depression, anxiety, increased suicidal risks, comorbid chronic conditions, and high frequency of accidents and falls \[[@B2]--[@B4]\]. Moreover, chronic sleep disturbance can seriously compromise the overall quality of life of those who suffer from it \[[@B5], [@B6]\]. Given the adverse effects of prolonged use of hypnotics, such as morning sedation, impaired balance, drug dependence, depression, and amnesia \[[@B1], [@B7]\], the exploration of noninvasive and nonpharmacological complementary methods for insomnia amongst the elderly is warranted. Auriculotherapy (AT) is a traditional Chinese medicine (TCM) approach in which the ear is viewed as a microsystem of the body \[[@B8]\]. AT is a therapeutic method where specific points on the auricle are stimulated to treat various bodily disorders. Different materials, such as acupuncture needles, press tack needles, seeds, magnetic pellets, or low-energy laser, could be applied on auricular points (denoted as "acupoints" in this paper) located on the external ear for therapeutic effect \[[@B8]--[@B11]\]. However, auricular acupuncture may induce patients\' discomfort and cause infection and inflammation in the puncture sites. Magneto-auriculotherapy (MAT) has gradually emerged as a popular intervention for treating many chronic problems, such as insomnia \[[@B12]\], low back pain \[[@B13]\], constipation \[[@B14]\], and hypertension \[[@B15]\]. The effectiveness of magnetic pellets may be attributed to the functional changes caused by the interaction of magnetic fields with biological tissues. Such changes may be related to moving ions in blood \[[@B16]\]. Laser auriculotherapy (LAT) has also been widely used in different medical conditions, including insomnia \[[@B17]\], pain relief \[[@B18]\], and weight reduction \[[@B19]\]. The combination of LAT and other treatments proposed in the literature produces a synergistic effect. LAT has been combined with ear point pressing to treat bed wetting in children \[[@B20]\] and with auricular pressing therapy for alcoholic addiction \[[@B21]\]. According to TCM, the laser beam irradiates and stimulates the acupoint and activates the therapeutic effects of*qi* (energy flow), thereby regulating the functions of*zang-fu* (internal organs) and restoring*yin-yang*(equilibrium) to produce a therapeutic effect \[[@B22]\]. Laser treatment is noninvasive, painless and presents no risk of infection or cross infection \[[@B22]\]. As such, the therapeutic benefits of laser combined with MAT merit further investigation. LAT followed by MAT optimises the therapeutic effect because the latter allows continuous stimulation of acupoints after the laser treatment, as long as the magnet pellets on the ears are in situ. In this study, three minimally invasive procedures, namely, LAT, MAT, and their combination, were investigated to determine the desirable treatment modality using AT to improve the sleep conditions of the elderly. Compared with the separate treatment procedures of MAT and LAT, their combined used is hypothesised to be more effective in improving the sleep conditions and thereby the quality of life of the elderly with insomnia. 2. Methods {#sec2} ========== 2.1. Design {#sec2.1} ----------- This study employed a three-arm double-blinded randomised trial. Eligible subjects were randomly divided into three groups by using a computer-generated randomised table and the equal proportion rule (1:1:1). The random coding was concealed from the subjects and evaluator by using opaque envelopes. ### 2.1.1. Settings and Participants {#sec2.1.1} Through convenience sampling, subjects were recruited from elderly centres in Hong Kong. A recruitment talk on AT was given to potential subjects in the targeted elderly centres. The definition of insomnia is adapted from existing literature \[[@B23]\]. After a preliminary screening, volunteers aged 65 years or above were recruited if they have the following symptoms: (1) difficulty falling or staying asleep and/or frequent nocturnal awakenings at least three nights per week, (2) sleep disturbance lasting for a minimum of 6 months, and (3) poor quality of sleep as indicated by a PSQI score greater than five. All the subjects fulfilled the criteria stipulated for the diagnosis of insomnia in the 'Diagnostic and Statistical Manual of Mental Disorders\', fifth edition \[[@B24]\]. The exclusion criteria were as follows: (1) presence of profound physical illnesses such as stroke, (2) diagnosis of obstructive sleep apnoea, (3) wearing a hearing aid or pacemaker in situ (to prevent the magnetic pellets from interfering with the devices), (4) received AT within the preceding six months, (5) suffering from aural injuries or infections, and (6) inability to understand instructions or provide consent. 2.2. Intervention and Procedures {#sec2.2} -------------------------------- ### 2.2.1. Acupoints Selection {#sec2.2.1} Seven auricular points, namely, "shenmen", "heart", "liver", "spleen", "kidney", "occiput", and "subcortex" ([Figure 1](#fig1){ref-type="fig"}), were selected because they promote sleep, as verified in a previous study by the first author \[[@B12]\]. The selection was based on the nomenclature and location of acupoints published by the China Standardisation Organising Committee (GB/T 13734-2008) \[[@B25]\]. Therapy was delivered by research personnel (SY) who had received intensive coaching from the first author (LS). Establishing the interrater reliability and accuracy of the ear point identification scheme ensured the fidelity of the study. ### 2.2.2. Groupings {#sec2.2.2} *Group 1 (Placebo LAT and MAT)*. The laser device was switched to "power off" mode (i.e., deactivated) for acupoint 'stimulation\' to achieve blinding and the placebo effect, before the MAT. The subjects were asked to wear a pair of laser-protective goggles to 'blind\' them during treatment. MAT was then applied by placing magnetic pellets on the selected acupoints ([Figure 2](#fig2){ref-type="fig"}). Each magnetic pellet has an average gauss/pellet magnetic flux density of approximately 200 Gs (20 mT) and a diameter of 1.76 mm. *Group 2 (LAT and Placebo MAT)*. A laser device (Pointer Pulse™) was used for LAT. The device has a wavelength of 650 nm, average output power of 2.5 mW, energy density of 0.54 J/cm^2^ for 1 minute, and a pulse of 10 Hz, which is a commonly acceptable dosage for clinical use \[[@B18], [@B26]\]. LAT used low-level laser therapy (LLLT), in which the energy level emitted from the device is comparable with that of the teaching pointer. The continuous mode of the device was used to directly treat the acupoints for one minute ([Figure 3](#fig3){ref-type="fig"}). A plaster centred with a small dried stem of*Junci medulla*, a soft perennial plant, was provided to mimic MAT. In a previous study,*J. medulla*was successfully adopted as a placebo because it did not induce any physical pressure on the acupoints of the ear \[[@B12]\]. *Group 3 (Combined AT)*. The subjects received the combined LAT and MAT. The procedures for applying LAT and MAT were identical to the abovementioned descriptions. ### 2.2.3. Procedures {#sec2.2.3} Therapies were administered at elderly centres adjacent to the subjects\' residences. The following procedures were standardised across the three groups to enhance the blinding effect. All therapies were administered in a room assigned for research purposes. Laser-protective goggles specific for the wavelengths of the laser device were provided to the subjects and researchers for eye protection. The auricle of every subject was cleaned with 75% isopropyl alcohol before therapy. Only one ear was treated at a time. The treatment was applied alternately to the right ear in the first visit and then to the left ear in the subsequent visit. We replaced the experimental objects (i.e., magnetic pellets for true MAT or*J. medulla* for placebo MAT) every other day, that is, three times a week (except Sunday) to prevent local irritation of acupoints. The total treatment period was six weeks. Participation in the study was voluntary. Written informed consent was obtained from each subject upon explaining the risks and benefits of their participation. Ethical approval was obtained from the Human Research Ethics Review Committee of the Hong Kong Polytechnic University. The study was conducted in accordance with the Declaration of Helsinki. Given their multiple visits to the centres to receive treatment, the subjects were provided a travel subsidy in the form of supermarket coupons upon completion of the study. ### 2.2.4. Treatment Effect Evaluation {#sec2.2.4} The subjects were assessed at baseline, at six weeks (postintervention), and during follow-up after six weeks, three months, and six months. To achieve evaluator blinding, the assessment was conducted by a different researcher who was unaware of the treatment modality given to the subjects. PSQI, which was used to collect data related to the sleep conditions of the subjects, was considered as the primary outcome. This instrument was scored from 0 to 21, and scores greater than five indicated poor sleep quality. Chong and Cheung \[[@B27]\] validated the Cantonese PSQI and reported a high internal consistency of 0.75. The secondary outcomes considered are as follows: (1) actigraphic monitoring was conducted to collect sleep parameters, including sleep latency (minutes), waking after sleep onset (minutes), total sleep time (hours), and sleep efficiency (%). An Actiwatch\' Spectrum Plus device with 0.025G ultra-high sensitivity and 32 Hz sampling rate was used in actigraphic monitoring. The subjects were requested to wear the device on the wrist of their nondominant hand 24 hours a day for 7 consecutive days to determine the overall sleep conditions within a certain period. Data were collected in epochs every 30 second. These epoch-by-epoch data were stored in the internal memory of the device until they could be downloaded to a computer. Actiware 6 Actigraph Analysis Software was used for sleep analysis. (2) The Chinese (HK) SF-12 v2©, an abbreviated version of the SF36 health questionnaire, was used to evaluate the health-related quality of life (HRQOL) of the subjects. This instrument covered 12 items, and the results were presented by a physical component score (PCS) and a mental component score (MCS). PCS and MCS ranged from 0 to 100, with higher scores indicating better HRQOL \[[@B28]\]. (3) Patient Health Questionnaire (PHQ-9) was also used. This instrument was validated as a useful tool for assessing depression status. The scores ranged from 0 to 27, and high scores indicated severe depression status \[[@B29]\]. The scale had a Cronbach\'s alpha of 0.82 and the recommended cut-off score was 8 \[[@B30]\]. The subjects\' expectations and satisfaction towards the therapy were evaluated using a 10-point scale, with high scores indicating high expectations or satisfaction \[[@B31]\]. Data were likewise collected on sociodemographic characteristics including age, gender, marital status, educational level, religion, number of family members, body mass index, single/shared bed, comorbid illnesses, use of sleeping pills or aids, and current medications taken. Similarly monitored were the recruitment rate, compliance rate of the treatment protocol, and adverse effects arising from the therapy. ### 2.2.5. Data Analyses {#sec2.2.5} Descriptive statistics were determined on the sociodemographic and clinical characteristics of the subjects. The estimated mean and standard error were computed for the outcome variables of each time point. Association amongst categorical variables was estimated using x^2^ test or Fisher\'s exact test, where appropriate, to identify significant variables for inclusion in the generalised estimating equations (GEE) for adjustment. One-way analysis of variance was used to examine group differences. Primary analysis was conducted using GEE model with an autoregression correlation structure to evaluate interactions amongst the groups over time (baseline to six months follow-up) on the primary outcome (i.e., PSQI score) and secondary outcomes (sleep parameters using 24-hour actigraphic monitoring, quality of life using SF-12, and PHQ-9). Missing data were handled using GEE and assumed to be random \[[@B32]\]. The main analysis was repeated at postintervention and during the follow-up sessions (up to six months) for sensitivity analysis. SPSS version 25.0 (IBM Corporation, USA) was used for all statistical analyses. All statistical tests were two sided with significance level set at 0.05. 3. Results {#sec3} ========== The study was conducted from May 2016 to May 2018. Data were collected from 11 centres for the elderly, and the recruitment rate was 88.6%. A total of 147 eligible subjects were randomly divided into three groups (Group 1=50, Group 2=46, and Group 3=51). 3.1. Participants\' Characteristics {#sec3.1} ----------------------------------- The recruited subjects had an average age of 75.29 years± 6.99, with a mean duration of insomnia for 10.12 ± 10.67 years. Majority of the subjects (70.0%) did not take any medication to manage their sleep problems. The groups were essentially comparable and well balanced in terms of sociodemographic variables, including gender distribution, body mass index, education level, marital status, comorbid illnesses, and regular medications taken. However, age showed slight significant differences amongst the groups, and thus this variable was adjusted in the GEE models in subsequent analyses. According to the 24-hour actigraphic recordings, the subjects had an average of poor sleep quality (PSQI 12.63 ±3.24, sleep efficiency 72.33% ±16.09%), long sleep latency (27.03 ± 23.13 minutes), short total sleep time (3.76 ± 1.96 hours), and waking after sleep onset (90.31 ± 88.53 minutes). The subjects also had mild depression (PHQ-9 9.47 ± 6.07) and low HRQOL in terms of physical component (41.39 ± 8.51) and mental component (46.68 ± 12.34) ([Table 1](#tab1){ref-type="table"}). 3.2. Compliance, Expectation, and Satisfaction towards the Treatment {#sec3.2} -------------------------------------------------------------------- Compliance with the intervention protocol was high, at an average of 95.2% (*n*= 140) of the subjects continued with postintervention and all follow-up measurements until six months. The recruitment flowchart is illustrated in [Figure 4](#fig4){ref-type="fig"}. Although majority of the subjects (65.3%) had never tried complementary and alternative treatments, they generally exhibited strong confidence in the proposed therapy (7.82 of 10) and had a relatively high expectation of its effectiveness (7.73 of 10) before the trial. After the intervention, the subjects in Group 1 had the highest satisfaction from the therapy (7.86), followed by those in Group 3 (7.58) and Group 2 (7.02). A correlation analysis was conducted between expectations of the treatment effect and sleep parameters (PSQI, SE). However, no significant relationship was detected (p\>0.05). Over 75% of the subjects (*n* = 109) indicated that they would definitely recommend the therapy to others. No specific adverse effects were observed arising from the therapy, apart from 16 cases (10.9%) who reported having mild skin irritation on the ears due to the adhesive tapes that were used to hold the experimental tools in place and 20 cases (13.6%) who felt tenderness on the acupoints (most of these subjects received MAT). The number of subjects who believed that they might be receiving placebo treatment was higher in Group 1 than in the other groups, although majority of the subjects (90%) believed that they were not receiving placebo ([Table 2](#tab2){ref-type="table"}) 3.3. Treatment Effect {#sec3.3} --------------------- The differences in the primary and secondary outcomes of the three groups across different time points were compared through GEE model analysis, with adjustment for age. In general, no significant differences were detected in the outcomes (including PSQI, sleep parameters measured by actigraphic monitoring, SF-12, and PHQ-9) of the three groups ([Table 3](#tab3){ref-type="table"}). However, significant differences were found in all of the subjective measures, including PSQI ([Figure 5](#fig5){ref-type="fig"}), SF-12 (physical and/or mental components) (Figures [S1](#supplementary-material-1){ref-type="supplementary-material"} and [S2](#supplementary-material-1){ref-type="supplementary-material"}), and PHQ-9 ([Figure 6](#fig6){ref-type="fig"}) for individual groups over time. When the sleep conditions were evaluated by actigraphic monitoring, significant differences in 'waking after sleep onset\' (minutes) ([Figure S3](#supplementary-material-1){ref-type="supplementary-material"}) and 'sleep efficiency\' (%) ([Figure S4](#supplementary-material-1){ref-type="supplementary-material"}) were detected only in Groups 1 and 3 ([Table 4](#tab4){ref-type="table"}). The completers\' analysis showed consistent findings on the primary and secondary outcomes of the trial. 4. Discussion {#sec4} ============= Numerous studies that used AT to manage sleep problems in China encountered methodological flaws, which rendered their findings unconvincing. The common problems included lack of details on how randomisation and allocation concealment were conducted, absence of objective measurements and a control or placebo group as well as failure to report the use of blinding and selective reporting of findings \[[@B33]--[@B35]\]. The present study was performed using a scientific approach to identify the optimum treatment protocol for AT in improving the sleep conditions and quality of life of the elderly with insomnia. This meticulous randomised controlled trial (RCT) could provide scientific evidence regarding causal relationships between interventions and outcomes. In general, the treatment effect was comparable amongst the three AT protocols. However, significant improvements were observed in all of the subjective measures, including sleep conditions measured by PSQI, HRQOL, and depression status for individual groups over time. When the sleep conditions were evaluated by objective measures using actigraphic monitoring, significant differences in "waking after sleep onset" (minutes) and "sleep efficiency" (%) were only noted in Groups 1 and 3. The use of actigraphs has been widely recognised as an objective measurement that could provide longitudinal assessment of sleep patterns in a natural environment \[[@B36], [@B37]\]. This technique could similarly provide valid measures that may not be influenced by subject bias. Significant reduction in the awakening time after sleep onset and increase in sleep efficiency were only detected in subjects who received MAT protocols (i.e., Groups 1 and 3) but not in those who received only the LAT protocol. Despite the improved sleep efficiency of the subjects at postintervention and during the follow-up periods, the evaluated indices remained below 85%, a common cut-off percentage that indicates the presence of sleep disturbances \[[@B38]\]. Sleep efficiency is calculated by dividing the total sleep time by total bedtime; a higher sleep efficiency means better sleep quantity and quality. A long therapeutic course, such as 10--12 weeks, may be necessary to further elevate sleep efficiency to a desirable level through sustainable treatment effect. MAT could provide continuous stimulation of acupoints as long as the magnetic pellets on the ears are in situ, and the subjects could receive laser stimulation to the acupoints on the day of treatment. The synergistic effect of the combined MAT and LAT was demonstrated in two previous trials \[[@B39], [@B40]\] conducted by the research team. In a double-blind RCT for osteoarthritic knee, the subjects who received the combined AT protocols exhibited stronger treatment effects in terms of pain relief, ambulation status, and range of knee movements compared with those treated with separate MAT or LAT \[[@B39]\], whereas in another double-blinded RCT for aging males with lower urinary tract symptoms, a combined AT protocol exhibited a stronger therapeutic effect in relieving voiding problems, improving the urinary flow rate, and minimizing the postvoid residual urine than the placebo group or MAT alone \[[@B40]\]. However the combined MAT and LAT approach did not show any advantage over the separate MAT protocol in current study. Therefore a greater frequency for LAT administration, such as daily application adopted in previous studies \[[@B41]--[@B44]\], may be considered in future studies to enhance the treatment effect and possibly improve its synergistic effect when combined with MAT. Numerous clinical trials reported the use of MAT on different disorders, including but not limited to sleep disturbances \[[@B12], [@B45], [@B46]\], low back pain \[[@B13]\], constipation \[[@B14]\], and obesity \[[@B47]\]. The effectiveness of MAT may relate to the interaction of magnetic fields with blood flow and calcium channel proteins in the cell membrane. Such interactions may elicit functional body changes \[[@B16], [@B48], [@B49]\]. Meanwhile, LAT is a noninvasive alternative to needle acupuncture \[[@B22]\] and has been adopted to increase pain threshold \[[@B50]\] as well as relieve musculoskeletal pain \[[@B18]\] and insomnia \[[@B17]\]. The laser beam not only irradiates and stimulates acupoints but also triggers the energy flow (*qi*) and regulates the functions of internal organs to achieve a therapeutic effect \[[@B22]\]. According to neuroembryonic theory, Dr. Paul Nogier viewed the auricle as a homunculus of the human body and has a similar shape to an inverted foetus \[[@B8]\]. As such, appropriate stimulation of specific ear acupoints can achieve therapeutic effects \[[@B51]\]. In the present study, the selection of auricular acupoints was based on the TCM theory and ideas borrowed from modern medicine. For example, treating the "heart" can calm the mind, while soothing the "liver" could regulate the flow of*qi*, particularly when insomnia is caused by liver*qi* stagnation \[[@B52]\]. "Shenmen" and "occiput" are believed to tranquilise the mind, and the 'subcortex\' can harmonise cortex excitement and inhibition \[[@B12]\]. A population-based epidemiological study conducted on 5,000 subjects in Hong Kong reported that insomnia was highly prevalent amongst Chinese adults and was associated with poor mental status and quality of life \[[@B53]\]. The present study reported PCS scores of the subjects comparable with those of Hong Kong people with insomnia (43.21 in our study versus 41.39 population norm), but stated a slightly higher MCS (46.68 versus 36.36). HRQOL (in PCS and MCS) improved and depression declined over time in subjects treated with AT. Thus, quality of life and emotional status of the subjects may be positively associated with sleep improvement after the therapy. No specific adverse effects were observed to arise from the therapy, apart from a small number of reported cases (10.9%) having mild skin irritation on the ears from the adhesive tapes used to hold the experimental tools in place. Moreover, 20 cases (13.6%) reported tenderness on the acupoints, and most of these subjects received MAT. According to the auricular diagnosis system, the areas of the auricle with heightened tenderness upon touching correspond to specific areas of the body where pathological conditions exist \[[@B54], [@B55]\]. Applying magnetic pellets may induce physical pressure on the ear acupoints and cause tenderness, especially in cases with disequilibrium of bodily functions (e.g., insomnia) corresponding to specific acupoints. The tenderness on the reflective acupoints experienced by the subjects may be considered part of the treatment process rather than adverse effects of AT. The high compliance rate (95.2%) and positive impression of the therapy indirectly indicated that blinding was successful because of the successful placebo application in the trial. Over 75% of the participants expressed that they would definitely recommend the therapy to others. The findings of this trial can provide valuable information and increase understanding of the therapeutic effect of AT, whether separate or combined MAT and LAT. Longer therapy duration may be considered in future trials to determine further improvements in the outcome variables. The proposed treatment approach can be considered as a noninvasive strategy for managing insomnia amongst the elderly. Although actigraphic monitoring is a reliable objective measure in sleep study, changes in different sleep stages caused by the therapy cannot be ascertained. The recruited elders generally obtained low education levels and had a mean age of over 75, and therefore, using a sleep log to verify the actigraphic data was not feasible. Night-to-night variability in the sleeping patterns of subjects may also be affected by physical, psychological, and/or environmental factors. Due to the above limitations, the actigraphic data must be interpreted with caution. The exact mechanisms remain unknown regarding the interaction of magnetic fields with biological tissues, which results in functional changes. Further studies from the biomedical perspective are required to elucidate the biological pathway of the treatment and effect, such as to examine the impact of the treatment protocols on the changes in sleep biomarkers when sleep prosperity is achieved. 5. Implications {#sec5} =============== This study provides valuable findings regarding the therapeutic effect of different protocols using MAT, LAT, or their combination. In general, the AT protocols under testing may be considered as a noninvasive approach with minimal adverse effects for managing sleep problems amongst the elderly. The findings of this study provide important implications to guide future research and apply evidence-based practice in the community via service provision to manage this common problem. 6. Conclusion {#sec6} ============= The treatment effects of the three protocols were comparable in terms of self-reported sleep conditions, HRQOL, and depression status. In several parameters, such therapeutic effects may be sustained at six-month follow-up. Significant improvement in the objective sleep parameters could be observed in subjects who received MAT protocols but not in those who received LAT. The combined MAT and LAT approach did not show any advantage over MAT. Longer therapeutic course and more frequent administration of LAT may be considered in future trials to achieve the optimal treatment effect. In general, the proposed AT protocols were demonstrated to be a well-received treatment modality with minimal adverse effects and effectively improved sleep conditions of the elderly with insomnia. The authors extend their appreciation to the elderly centres and participants for their sincere support for this study. This project was supported by the Health and Medical Research Fund, Food and Health Bureau, Hong Kong SAR Government (\#13144061). Data Availability ================= The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request Ethical Approval ================ Ethical approval from the Human Research Ethics Review Committee of the Hong Kong Polytechnic University was granted \[Reference no. HSEARS20151129001\]. Participation in the study was on a voluntary basis. All potential participants were assured that they could withdraw from the study at any time. Personal information and data remained confidential and anonymous. Consent ======= Written informed consent was obtained from each participant upon recruitment. Conflicts of Interest ===================== The authors declare that they have no conflicts of interest. Authors\' Contributions ======================= Lorna K. P. Suen was the principal investigator. Lorna K. P. Suen, A. Molassiotis, and C. H. Yeh were involved in conception and design of the study. S. K. W. Yueng collected the data. Lorna K. P. Suen was responsible for data analysis. Lorna K. P. Suen drafted the manuscript with the support of all authors. All authors read and approved the final manuscript. Supplementary Materials {#supplementary-material-1} ======================= ###### Figure S1: wake after sleep onset (minutes) across time. Figure S2: sleep efficiency (%) across time. Figure S3: SF-12 (physical component) across time. Figure S4: SF-12 (mental component) across time. ###### Click here for additional data file. {#fig1} {#fig2} {#fig3} {#fig4} {#fig5} {#fig6} ###### Sociodemographic and baseline characteristics of the participants sample (*N*=147). ------------------------------------------------------------------------------------------------------------------------   All\ Group 1\ Group 2\ Group 3\ *P*-value (*N*=147) Placebo LAT & MAT\ LAT & placebo MAT\ Combined AT\ (*n*=50) (*n*=46) (*n*=51) ---------------------------------- --------------- -------------------- -------------------- --------------- ----------- Age (years)         ❖0.025*∗* Mean (SD) 75.29 (6.99) 76.02 (7.02) 76.80 (6.16) 73.20 (7.29)   Gender           Male 19 7 5 7 ★ 0.907 Female 128 43 41 44   Education level           Primary or below 100 32 35 33 *◈*0.734 Secondary 39 15 9 15   Tertiary or above 8 3 2 3   Marital status           Single 5 2 1 2 *◈*0.971 Married 82 29 26 27   Divorced/Widowed 60 19 19 22   Religion           No 75 26 22 27 ★ 0.895 Yes 72 24 24 24   Body mass index (kg/m^2^) 22.29 (3.73) 22.85 (3.48) 21.50 (3.57) 22.46 (4.04) ❖ 0.076 Shared bed           No 109 33 38 38 ★ 0.188 Yes 38 17 8 13   Living alone           No 66 20 27 19 ★ 0.079 Yes 81 30 19 32 Sleeping pills taken           No 103 34 33 36 *◈*0.391 Previous user 32 14 7 11   Current user 12 2 6 4   Duration of insomnia (years)\ 10.12 (10.67) 9.23 (8.34) 11.15 (13.09) 10.06 (10.40) ❖ 0.678 Mean (SD) Comorbid illness           No 22 5 5 12 ★ 0.488 Yes 125 45 41 39   Regular drugs taken           No 32 10 7 15 ★ 0.441 Yes 115 40 39 36   PSQI (total) (0-21) 12.63 (3.24) 12.67 (3.15) 12.89 (4.13) 12.35 (2.34) ❖ 0.715 Sleep latency (min) 27.03 (23.13) 26.56 (23.76) 28.68 (24.86) 26.00 (21.19) ❖ 0.839 Total sleep time (hours) 3.76 (1.96) 3.93 (1.93) 3.74 (1.86) 3.62 (2.10) ❖ 0.734 Wake after sleep onset (minutes) 90.31 (88.53) 92.06 (91.44) 90.03 (99.56) 88.86 (75.95) ❖ 0.984 Sleep efficiency (%) 72.33 (16.09) 72.44 (15.91) 74.25 (16.04) 70.50 (16.42) ❖ 0.521 SF-12 (PCS) 41.39 (8.51) 40.65 (7.58) 41.43 (9.69) 42.07 (8.36) ❖ 0.707 SF-12 (MCS) 46.68 (12.34) 48.22 (11.76) 46.19 (13.49) 45.62 (11.89) ❖ 0.542 PHQ-9 9.47 (6.07) 9.54 (5.34) 9.52 (6.69) 9.35 (6.26) ❖ 0.986 ------------------------------------------------------------------------------------------------------------------------ SD, standard deviation. PSQI: Pittsburgh sleep quality index. SF-12 (PCS): physical component score. SF-12 (MCS): mental component score. PHQ-9: Patient Health Questionnaire, for depression. ❖One-way analysis of variance. *◈*Fisher\'s exact test. ★Chi-square test. *∗*Statistically significant at *P* \< 0.05. ###### Reported adverse effects, expectations, and satisfaction towards the therapy ¥. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------   All\ Group 1\ Group 2\ Group 3\ *P-*value (*n*=147) Placebo LAT & MAT\ LAT & placebo MAT\ Combined AT\ (*n*=50) (*n*=46) (*n*=51) ------------------------------------------------------------------- ------------- ------------------------- ------------------------- ------------------------- ------------- Have you used complementary therapies in the past?\#         ★ 0.423 No 96 (65.3%) 33 33 30   Yes 51 (34.7%) 17 13 21   How much faith do you have in complementary therapies in general\ 7.82 (2.13) 8.40 (1.94) 7.48 (2.18) 7.55 (2.18) ❖ 0.056 (0 to 10) \# Expectation for treatment effect towards MAT\ 7.41 (2.10) 7.96 (2.11) 6.93 (2.03) 7.31 (2.07) ❖ 0.051 (0 to 10) \# Expectation for treatment effect towards LAT\ 7.41 (2.13) 8.08 (2.00) 6.87 (2.09) 7.25 (2.15) ❖ 0.016*∗* (0 to 10) \# Average expectation for treatment effect\ 7.73 (2.13) 8.40 (2.03) 7.20 (2.03) 7.57 (2.18) ❖ 0.016*∗* (0 to 10) \# Ear itchiness ¥ 16 (10.9%) 8*∗*  \ 2*∗*  \ 6*∗*  \ - - - (resolve automatically) (resolve automatically) (resolve automatically) Tenderness on acupoints ¥ 20 (13.6%) 9*∗*  \ 1*∗*  \ 10*∗*  \ - - - (resolve automatically) (resolve automatically) (resolve automatically) Satisfaction towards therapy (0 to 10) ¥ 7.58 (2.37) 7.86 (2.03) 7.02 (2.68) 7.58 (2.37) ❖ 0.225 Thought that they might be receiving placebo treatment ¥           No 126 (90.0%) 39 41 46 *◈*0.011*∗* Yes 14 (10.0%) 10 3 1   Will recommend this therapy to others ¥         *◈* 0.166 Definitely will 109 (75.7%) 38 29 42   Maybe 20 (13.9%) 5 9 6   No 15 (10.4%) 6 7 2   ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Mean (standard deviation) unless otherwise noted. *Ω*Association between variables was determined by chi-square analyses or Fisher\'s-exact test where appropriate. \#Evaluated before the intervention. ¥Evaluated after the intervention has been completed. *∗*"Certain" causality. ❖One-way analysis of variance. *◈*Fisher\'s exact test. ★Chi-square test. *∗*Statistically significant at *P*\<0.05. ###### Outcome variables across three groups at different timepoints.         Pairwise comparisons between groups *▾* ------------------------------------ --------------- --------------- --------------- ----------------------------------------- ------------------------- ----------------------- *PSQI (total) (0-21)*             Baseline 12.68 (0.44) 12.90 (0.62) 12.34 (0.34)       Post intervention 9.46 (0.52) 9.91 (0.55) 8.48 (0.51) 0.23 (-1.44, 1.89) -0.65 (-2.36, 1.06) 0.87 (-0.55, 2.30) 6 weeks follow up 9.07 (0.53) 9.25 (0.65) 9.26 (0.54) -0.50 (-2.07, 1.97) 0.53 (-1.14, 2.19) -0.58 (-2.54, 1.38) 3 months follow up 9.12 (0.79) 7.81 (0.68) 9.06 (0.58) -1.54 (-4.00, 0.93) 0.27 (-1.88, 2.42) -1.81 (-3.99, 0.37) 6-month follow up 8.20 (0.62) 7.96 (0.67) 8.46 (0.66) -0.47 (-2.68, 1.75) 0.59 (-1.38, 2.56) -1.06 (-3.16, 1.05) *Sleep latency (minutes)*             Baseline 26.82 (3.33) 29.16 (3.59) 25.17 (2.86)       Post intervention 25.29 (2.97) 25.00 (3.32) 26.17 (3.03) (-13.29, 8.02) 2.23 (-8.11, 12.57) -4.86 (-15.33, 5.60) 6 weeks follow up 19.35 (3.22) 26.56 (3.39) 23.68 (3.30) 4.87 (-7.73, 17.47) 5.67 (-5.89, 17.23) -0.80 (-11.77, 10.18) 3 months follow up 17.99 (3.37) 21.29 (3.56) 23.90 (4.22) 0.96 (-11.49, 13.41) 7.25 (-6.93, 21.43) -6.29 (-20.65, 8.07) 6-month follow up 19.66 (5.31) 23.42 (5.39) 19.07 (4.34) 1.43 (-15.79, 18.64) 0.76 (-15.34, 16.86) 0.67 (-15.85, 17.18) *Total sleep time (hours)*             Baseline 3.93 (0.27) 3.75 (0.28) 3.61 (0.30)       Post intervention 3.93 (2.49) 4.05 (0.30) 3.96 (0.29) 0.31 (-0.67, 1.28) 0.35 (-0.65, 1.36) -0.05 (-1.17, 1.08) 6 weeks follow up 3.81 (0.27) 3.90 (0.29) 3.85 (0.25) 0.27 (0.68, 1.23) 0.36 (-0.59, 1.31) -0.09 (-1.16, 0.99) 3 months follow up 3.66 (0.44) 4.26 (0.49) 3.38 (0.24) 0.77 (-0.81, 2.36) 0.034 (-1.32, 1.39) 0.74 (-0.61, 2.09) 6-month follow up 3.92 (0.49) 4.42 (0.61) 3.71 (0.45) 0.68 (-1.03, 2.38) 0.112 (-1.45, 1.68) 0.57 (-1.14, 2.27) *Wake after sleep onset (minutes)*             Baseline 92.32 (12.83) 90.51 (14.64) 88.33 (10.31)       Post intervention 69.44 (10.41) 71.60 (9.49) 65.84 (8.86) 3.97 (-22.03, 29.96) 0.38 (-23.41, 24.18) 3.58 (-24.42, 31.59) 6 weeks follow up 51.45 (5.10) 61.40 (8.60) 62.34 (7.07) 11.76 (-24.44, 47.95) 14.88 (-17.49, 47.25) -3.13 (-39.79, 33.54) 3 months follow up 50.82 (9.13) 74.74 (20.96) 63.90 (12.00) 25.73 (-38.06, 89.51) 17.06 (-30.43, 64.55) 8.67 (-55.72, 73.06) 6-month follow up 48.81 (9.66) 80.87 (26.46) 61.59 (11.04) 33.87 (-38.57, 106.30) 16.77 (-30.71, 64.24) 17.10 (-55.55, 89.75) *Sleep efficiency (%)*             Baseline 72.34 (2.23) 74.06 (2.38) 70.70 (2.23)       Post intervention 78.28 (1.90) 76.94 (1.93) 78.86 (1.64 ) -3.06 (-8.52, 2.40) 2.22 (-4.06, 8.50) -5.28 (-11.61, 1.05) 6 weeks follow up 77.57 (1.80) 76.14 (2.49) 74.57 (2.21) -3.14 (-9.31, 3.03) -1.36 (-7.71, 5.00) -1.78 (-8.75, 5.18) 12-week follow up 81.32 (1.98) 78.02 (3.56) 78.82 (2.34) -5.03 (-15.44, 5.39) -0.87 (-10.27, 8.54) -4.16 (-15.43, 7.10) 6-month follow up 76.02 (4.84) 77.37 (4.64) 80.19 (2.20) -0.37 (-14.64, 13.90) 5.81 (-6.18, 17.80) -6.18 (-18.97, 6.62) *SF-12 (PCS)*             Baseline 40.87 (1.09) 41.84 (1.37) 41.64 (1.19) \-     Post intervention 44.46 (1.23) 43.12 (1.38) 42.96 (1.22) 2.31 (-6.00, 1.39) -2.27 (-5.69, 1.14) -0.035 (-3.62, 3.55) 6-weeks follow up 45.04 (1.34) 42.58 (1.44) 40.33 (1.51) -3.42 (-7.51, 0.66) -5.47 (-9.89, -1.06)*∗* 2.05 (-2.06, 6.15) 3 months follow up 45.55 (1.34) 46.21 (1.42) 41.30 (1.66) -0.31 (-4.69, 4.07) -5.02 (-9.53, -0.52)*∗* 4.72 (0.13, 9.30)*∗* 6-month follow up 46.26 (1.36) 45.96 (1.41) 41.99 (2.29) -1.26 (-5.63, 3.11) -5.03 (-10.55, 0.48) 3.77 (-2.06, 9.60) *SF-12 (MCS)*             Baseline 47.99 (1.63) 45.77 (1.95) 46.05 (1.63)       Post intervention 51.84 (1.64) 47.35 (1.78) 50.58 (.1.37) -2.26 (-7.10, 2.57) 0.675 (-4.02, 5.37) -2.94 (-7.42, 1.54) 6 weeks follow up 50.97 (1.61) 50.89 (1.84) 53.15 (1.65) 2.14 (-2.62, 6.90) 4.12 (-0.95, 9.18) -1.98 (-6.73, 2.77) 3 months follow up 48.05 (2.31) 51.31 (2.27) 50.55 (1.76) 5.48 (-1.52, 12.47) 4.44 (-1.80, 10.67) 1.04 (-5.38, 7.47) 6-month follow up 51.64 (2.00) 52.24 (2.05) 51.05 (2.20) 2.82 (-3.83, 9.47) 1.35 (-5.24, 7.94) 1.47 (-5.41, 8.36) *PHQ-9 (0-27)*             Baseline 9.52 (0.74) 9.48 (0.96) 9.40 (0.86)       Post intervention 6.30 (0.71) 6.77 (0.95) 6.46 (0.79) 0.504 (-1.52, 2.52) 0.28 (-1.78, 2.35) 0.22 (-1.72, 2.17) 6 weeks follow up 6.87 (1.00) 6.49 (1.11) 5.00 (0.71) -0.35 (-2.73, 2.04) -1.75 (-4.29, 0.78) 1.41 (-0.64, 3.45) 3 months follow up 6.58 (1.20) 5.68 (1.29) 6.55 (1.19) \--0.87 (-3.93, 2.21) 0.09 (-3.08, 3.26) -0.96 (-4.05, 2.14) 6-month follow up 4.77 (0.97) 5.82 (1.30) 5.68 (1.05) 1.10 (-1.88, 4.01) 1.04 (-1.87, 3.95) 0.06 (-3.07, 3.18) *※*Estimated mean and standard error (SE) from generalized estimating equations. PSQI: Pittsburgh sleep quality index. SF-12 (PCS): physical component score. SF-12 (MCS): mental component score. PHQ-9: Patient Health Questionnaire, for depression. *▾*Adjusted for age. *∗*Statistically significant at *P*\< 0.05. ###### Change in different outcome variables over time for individual groups using generalized estimating equations. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------   Post-hoc analyses *◈▾* ---------------------------------- ------------------------ ------------ ------------------------- ------------ ------------------------ ------------ ------------------------ ------------ Group 1\                 Placebo LAT & MAT\ (*n*=50) PSQI (0-21, total) -3.21 (-4.56, -1.87) 0.000*∗∗∗* -3.61 (-4.83, -2.38) 0.000*∗∗∗* -3.56 (-5.28, -1.83) 0.000*∗∗∗* -4.47 (-5.95. -3.00) 0.000*∗∗∗* Sleep latency (minutes) -1.53 (-8.97, 5.92) 0.688 -7.47 (-16.74, 1.81) 0.115 -8.83 (-17.48, -0.18) 0.045*∗* -7.16 (-19.08, 4.75) 0.239 Total sleep time (hours) -0.01 (-0.59, 0.58) 0.989 -0.12 (-0.69, 0.45) 0.681 -0.27 (-1.39, 0.86) 0.642 -0.01 (-1.12, 1.09) 0.985 Wake after sleep onset (minutes) -22.87 (-38.01, -7.74) 0.003*∗∗* -40.87 (-63.38, -18.36) 0.000*∗∗∗* -41.49 (-74,47, -8.52) 0.014*∗* -43.51 (-77.26, -9.76) 0.012*∗* Sleep efficiency (%) 5.94 (2.12, 9.76) 0.002*∗∗* 5.22 (1.36, 9.09) 0.008*∗∗* 8.98 (3.07, 14.90) 0.003*∗∗* 3.68 (-5.99, 13.35) 0.455 SF12 (PCS) 3.59 (1.09, 6.09) 0.005*∗∗* 4.16 (1.06, 7.27) 0.009*∗∗* 4.68 (1.64, 7.72) 0.003*∗∗* 5.38 (2.58, 8.19) 0.000*∗∗∗* SF12 (MCS) 3.85 (0.29, 7.41) 0.034*∗* 2.98 (0.61, 6.57) 0.103 0.06 (-4.76, 4.88) 0.980 3.65 (-0.83, 8.12) 0.110 PHQ-9 (0-27) -3.22 (-4.73, -1.71) 0.000*∗∗∗* -2.65 (-4.63, -0.66) 0.009*∗∗* -2.94 (-5.17, -0.71) 0.010*∗* -4.75 (-6.69, -2.82) 0.000*∗∗∗* Group 2\                 LAT & placebo MAT\ (*n*=46) PSQI (0-21, total) -2.99 (-3.96, -2.02) 0.000*∗∗∗* -3.66 (-5.26, -2.05) 0.000*∗∗∗* -5.09 (-6.85, -3.33) 0.000*∗∗∗* -4.94 (-6.60, -3.28) 0.000*∗∗∗* Sleep latency (minutes) -4.16 (-11.78, 3.46) 0.284 -2.59 (-11.15, 5.96) 0.552 -7.87 (-16.82, 1.08) 0.085 -5,74 (-18.18, 6.71) 0.366 Total sleep time (hours) 0.30 (-0.47, 1.08) 0.444 0.15 (-0.61, 0.92) 0.695 0.51 (-0.61, 1.62) 0.375 0.67 (-0.64, 1.97) 0.316 Wake after sleep onset (minutes) -18.91 (-40.04, 2.23) 0.080 -29.11 (-57.49, -0.72) 0.044*∗* -15.76 (-70.40, 38.88) 0.572 -9.64 (-74.21, 54.94) 0.770 Sleep efficiency (%) 2.88 (-1.03, 6.78) 0.148 2.08 (-2.72, 6.89) 0.396 3.96 (-4.62, 12.53) 0.366 3.31 (-7.32, 13.95) 0.542 SF12 (PCS) 1.28 (-1.44, 4.00) 0.357 0.74 (-1.91, 3.39) 0.585 4.37 (1.21, 7.53) 0.007*∗∗* 4.12 (0.75, 7.49) 0.017*∗* SF12 (MCS) 1.58 (-1.68, 4.85) 0.341 5.12 (1.99, 8.25) 0.001*∗∗* 5.54 (0.48, 10.61) 0.032*∗* 6.47 (1.55, 11.39) 0.010*∗* PHQ-9 (0-27) -2.72 (-4.06, -1.38) 0.000*∗∗∗* -2.99 (-4.31, -1.67) 0.000*∗∗∗* -3.80 (-5.92, -1.69) 0.000*∗∗∗* -3.66 (-5.91, -1.41) 0.001*∗∗* Group 3\                 Combined AT\ (*n*=51) PSQI (0-21, total) -3.86 (-4.91, -2.82) 0.000*∗∗∗* -3.08 (-4.20, -1.95) 0.000*∗∗∗* -3.28 (-4.57, -2.00) 0.000*∗∗∗* -3.88 (-5.18. 2.58) 0.000*∗∗∗* Sleep latency (minutes) 0.70 (-6.47, 7.88) 0.848 -1.80 (-8.70, 5.10) 0.610 -1.58 (-12.82, 9.67) 0.784 -6.40 (-17.34, 4.54) 0.251 Total sleep time (hours) 0.35 (-0.46, 1.16) 0.399 0.24 (-0.52, 1.00) 0.535 -0.23 (-0.99, 0.52) 0.543 0.10 (-1.00, 1.21) 0.858 Wake after sleep onset (minutes) -22.49 (-40.86, -4.12) 0.016*∗* -25.98 (-49.25, -2.72) 0.029*∗* -24.43 (-58.62, 9.76) 0.161 -26.74 (-60.49, 7.01) 0.120 Sleep efficiency (%) 8.16 (1.18, 13.14) 0.001*∗∗* 3.87 (-1.18, 8.91) 0.133 8.12 (0.80, 15.44) 0.030*∗* 9.49 (2.31, 16,68) 0.010*∗* SF12 (PCS) 1.32 (-1.01, 3.64) 0.268 -1.31 (-4.45, 1.82) 0.413 -0.35 (-3.67, 2.98) 0.839 0.35 (-4.40, 5.10) 0.886 SF12 (MCS) 4.52 (1.46, 7.59) 0.004*∗∗* 7.10 (3.52, 10.67) 0.000*∗∗∗* 4.50 (0.55, 8.45) 0.026*∗* 5.00 (0.18, 9.82) 0.042*∗* PHQ-9 (0-27) -2.94 (-4.35, -1.53) 0.000*∗∗∗* -4.40 (-5.97, -2.83) 0.000*∗∗∗* -2.85 (-5.10, -0.59) 0.013*∗* -3.71 (-5.89, -1.54) 0.001*∗∗* ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- SE, standard error; CI, confidence interval. *◈*Timepoints: (1) = baseline, (2) = postintervention, (3) = 6-week follow up, (4) = 3-month follow up, and (5) = 6-month follow-up. *▾*Adjusted for age. PSQI = Pittsburgh sleep quality index. SF-12 (PCS): physical component score. SF-12 (MCS): mental component score. PHQ-9: Patient Health Questionnaire. *∗*Statistically significant at *P*\< 0.05. *∗∗* Statistically significant at *P*\<0.01. *∗∗∗* Statistically significant at *P*\<0.001. [^1]: Academic Editor: Mario Ledda

Q: python referencing json array all, I have a script I am building to find all open pull requests and compare the sha hash, however I can't seem to find them.... for repo in g.get_user().get_repos(): print (repo.full_name) json_pulls = requests.get('https://api.github.com/repos/' + repo.full_name + '/pulls?state=open+updated=<' + str(cutoff_date.date())+ '&sort=created&order=asc') if (json_pulls.ok): for item in json_pulls.json(): for c in item.items(): #print(c["0"]["title"]) #print (json.dumps(state)) print(c) The code cycles through the existing repos and list the pull requests and I get the output: But, I can't for the life of me figure out how to collect individual fields... I tried using the references: print(c['title']) - not defined is the error print(c['0']['title']) -a tubular error what I am looking for is a simple list for each request.... title id state base / sha head / sha Can someone please point out what I am doing wrong in referencing the json items in my python script as it is driving me crazy. The full code as is with your help of course ... : # py -m pip install <module> to install the imported modules below. # # # Import stuff from github import Github from datetime import datetime, timedelta import requests import json import simplejson # # #declare stuff # set the past days to search in the range PAST = 5 # get the cut off date for the repos 10 days ago cutoff_date = datetime.now() - timedelta(days=PAST) #print (cutoff_date.date()) # Repo oauth key for my repo OAUTH_KEY = "(get from github personal keys)" # set base URL for API query BASE_URL = 'https://api.github.com/repos/' # # # BEGIN STUFF # First create a Github instance: g = Github(login_or_token=OAUTH_KEY, per_page=100) # get all repositories for my account that are open and updated in the last no. of days.... for repo in g.get_user().get_repos(): print (repo.full_name) json_pulls = requests.get('https://api.github.com/repos/' + repo.full_name + '/pulls?state=open+updated=<' + str(cutoff_date.date())+ '&sort=created&order=asc') if (json_pulls.ok): for item in json_pulls.json(): print(item['title'], item['id'], item['state'], item['base']['sha'], item['head']['sha']) The repo site is a simple site, with two repos, and 1 or 2 pull requests to play against. The idea of the script, when it is done, it to cycle through all the repos, find the pull requests that are older than x days and open, locates the sha for the branch (and sha for the master branches, to skip..... ) remove the branches that are not master branches, thus removing old code and pull requests to keep the repos tidy.... A: json_pulls.json() returns a list of dictionaries, so you can just do: for item in json_pulls.json(): print (item['title'], item['id'], item['state'], item['base']['sha'], item['head']['sha']) There's no need to iterate over item.items().

Race/ethnicity, income, chronic asthma, and mental health: a cross-sectional study using the behavioral risk factor surveillance system. To examine the relationships among race/ethnicity, income, and asthma on mental health outcomes in individuals surveyed as part of the Centers for Disease Control and Prevention 2004 Behavioral Risk Factor Surveillance System (BRFSS). Racial and ethnic disparities in asthma prevalence exist, which may be explained in part by socioeconomic status. Individuals with asthma often have comorbid mental health conditions, the rates of which are also marked by significant racial and ethnic disparities. We obtained 2004 BRFSS demographic, asthma, and mental health data on Hispanics, non-Hispanic Whites, and non-Hispanic Blacks. Linear regression analysis was used to examine the main and interaction effects of race/ethnicity, income, and history of asthma on poor mental health (n = 282,011), as well as on depression (n = 14,907) and anxiety (n = 14,871) specifically. A significant three-way interaction emerged among race/ethnicity, income, and history of chronic asthma on number of days of poor mental health. Among the most impoverished (income <$15,000/yr), Hispanics with asthma reported greater number of days of poor mental health than non-Hispanic Whites with asthma. However, among those with slightly greater economic resources, Hispanics with asthma reported fewer number of days of poor mental health than non-Hispanic Whites. The results of this study highlight the complex interactions among race/ethnicity, income, and asthma on mental health outcomes.

Effects of Atorvastatin on Transient Sodium Currents in Rat Normal, Simulated Ischemia, and Reperfusion Ventricular Myocytes. (1) To detect the whether the effects of simulated ischemia on INa of rat left ventricular myocytes in a time-dependent manner and the effects of atorvastatin on ischemia INa; (2) To investigate the effects of atorvastatin on INa of rat-simulated ischemia/reperfusion (I/R) ventricular cells. Ventricular cells were enzymatically isolated by Langendorff perfusion system. Whole-cell patch clamp was applied to detect INa level. Some elements of extracellular fluid were hanged to simulate the status of normal, I and R condition. Then the effects of atorvastatin on INa were observed. (1) During simulated reperfusion, INa decreased and atorvastatin further suppressed the reduction degree. (2) At test potential -40 mV, no difference was detected among peak INa amplitude of ischemia for 20 min, reperfusion phase 3/5/7/9 min in continuous ischemia (I) group (p = 0.275). In I/R group, peak INa amplitude continuously decreased at 3 min (p = 0.005) and 9 min (p = 0.041). In atorvastatin intervention + I/R (Statin + I/R) group, peak INa amplitude at reperfusion 3 min decreased compared with ischemia phase (p = 0.000), while no significant difference was detected between 3 and 9 min (p = 0.858). The differences were significant at the same time point between groups. At reperfusion 3/5/7/9 min, peak INa of the I/R group was lower than the ischemia group (all p = 0.000), same as the Statin + I/R group (p = 0.000, p = 0.003, p = 0.006, and p = 0.001). Peak INa of the Statin + I/R group was higher than the I/R group at the same time point (p = 0.011, p = 0.033, p = 0.003, p = 0.003). There was no change in the I group during reperfusion phase (p > 0.05). In I/R group, V1/2 (mV) shifted from -58.87 ± 3.36 to -54.33 ± 2.40, k (mV) shifted from 1.25 ± 0.59 to 1.91 ± 0.84 (p < 0.05). In the Statin + I/R group, V1/2 (mV) increased from -57.80 ± 2.97 to -52.76 ± 3.14 (p < 0.01), no change was observed in k (p > 0.05). (1) In the status of reperfusion, INa decreased more than that in the status of ischemia. (2) Atorvastatin protected the cells from reduction of INa during long-time simulated (>15 min) I/R. (3) Overall, atorvastatin affected INa of the normal, simulated ischemic/reperfusion cell in rat left ventricle by blocking sodium channel -directly.

537 F.Supp. 1387 (1982) Vaughn E. PERRY, Jr., Plaintiff, v. The HARTZ MOUNTAIN CORPORATION, Defendant. No. IP 81-631-C. United States District Court, S. D. Indiana, Indianapolis Division. April 28, 1982. *1388 C. Robert Knight, Samuel A. Fuller, Indianapolis, Ind., for plaintiff. Gregory B. Craig, Washington, D. C., Joe C. Emerson, Indianapolis, Ind., for defendant. ENTRY DILLIN, District Judge. This case is before the Court on the motion to dismiss of the defendant, Hartz Mountain Corporation (Hartz). For the reasons stated below, the motion is granted in part and denied in part. Facts The plaintiff, Vaughn E. Perry, Jr., was employed by Hartz from September 1975 to June 14, 1979, when he was discharged. Perry accuses Hartz of certain anticompetitive practices. He alleges that his dismissal was in retaliation for his refusal to continue his participation in those practices and deliver evidence of them to Hartz. Hartz contends that Perry was fired because he defrauded the corporation and refused to cooperate in an in-house investigation of company practices. Perry filed suit on June 12, 1981, properly invoking jurisdiction under 28 U.S.C. §§ 1331 and 1332. He alleges (1) that he was wrongfully discharged, (2) that Hartz violated state and federal antitrust laws, (3) that Hartz negligently breached its duty of good faith and fair dealing, (4) that Hartz was guilty of outrageous conduct toward him, and (5) that Hartz defamed him. Hartz has moved to dismiss each of these claims for failure to state a claim upon which relief can be granted. Discussion I. Wrongful discharge Hartz contends that Perry has failed to state a claim for wrongful discharge because he was an employee at will who could be discharged at any time. Indiana has long subscribed to the employment at will doctrine, which holds that an employment at will relationship can be terminated at any time by either party. Speeder Cycle Co. v. Teeter, 18 Ind.App. 474, 48 N.E. 595 (1897). A modification of the doctrine occurred fairly recently in Frampton v. Central Indiana Gas Co., 260 Ind. 249, 297 N.E.2d 425 (1973). In that case the Indiana Supreme Court recognized an exception to the employment at will doctrine and allowed a cause of action for wrongful discharge by an employee who was dismissed in retaliation for exercising a statutorily conferred right — filing a workmen's compensation claim. *1389 In a case construing the Frampton rule, Campbell v. Eli Lilly & Co., Ind.App., 413 N.E.2d 1054 (1980), transfer denied, 421 N.E.2d 1099 (Ind.1981), the Indiana Court of Appeals equated exercising a statutorily conferred right with fulfilling a statutorily imposed duty, and stated that discharge in retaliation for either falls within the exception. However, the Court of Appeals found that the plaintiff, who charged that he was fired for reporting misconduct in drug research to company officials, had failed to establish either that he had exercised a statutory right or fulfilled a statutory duty. Id., 413 N.E.2d at 1059. In this case, on the other hand, Perry is under a statutory duty to refrain from engaging in conspiracies in restraint of trade. 15 U.S.C. § 1; I.C. 24-1-2-1. By alleging that Hartz discharged him for refusing to continue his participation in an anticompetitive conspiracy, Perry has stated a claim which falls within the Frampton-Campbell exception to the employment at will doctrine. On the wrongful discharge claim, therefore, Hartz's motion to dismiss is denied. II. Federal and state antitrust violations In Counts II and III of his complaint, plaintiff alleges that Hartz violated state and federal antitrust laws by, among other things, inducing retailers to deal exclusively with Hartz through payoffs and fraudulent credits and attempting to establish tying arrangements between Hartz's pet care and carpet care products. Hartz responds that Perry has no standing to assert these antitrust claims. Section 4 of the Clayton Act, 15 U.S.C. § 15, allows "[a]ny person who shall be injured in his business or property by reason of anything forbidden in the antitrust laws" to bring a private action. A literal interpretation of this section could lead to a flood of litigation by plaintiffs only indirectly affected by anticompetitive activities. Courts have approached the statutory language in various ways, seeking to make the right-to-sue standard manageable, yet still consistent with the purposes of the antitrust laws. See, e.g., Bravman v. Bassett Furniture Industries, Inc., 552 F.2d 90 (3rd Cir.), cert. denied, 434 U.S. 823, 98 S.Ct. 69, 54 L.Ed.2d 80 (1977) (balancing test); Malamud v. Sinclair Oil Corp., 521 F.2d 1142 (6th Cir. 1975) (zone of interests test); Reibert v. Atlantic Richfield Co., 471 F.2d 727 (10th Cir.), cert. denied, 411 U.S. 938, 93 S.Ct. 1900, 36 L.Ed.2d 399 (1973) (direct injury test); Mulvey v. Samuel Goldwyn Productions, 433 F.2d 1073 (9th Cir. 1970), cert. denied, 402 U.S. 923, 91 S.Ct. 1377, 28 L.Ed.2d 662 (1971) (reasonable foreseeability test); Karseal Corp. v. Richfield Oil Corp., 221 F.2d 358 (9th Cir. 1955) (target area test). In the Seventh Circuit the prevailing approach is the target area test, which provides that to state an antitrust claim a plaintiff must allege injury which is within the area affected, or intended to be affected, by the defendant's anticompetitive actions. Illinois v. Ampress Brick Co., 536 F.2d 1163, 1167 (7th Cir. 1976), rev'd on other grounds sub nom. Illinois Brick Co. v. Illinois, 431 U.S. 720, 97 S.Ct. 2061, 52 L.Ed.2d 707 (1977); In re Folding Carton Antitrust Litigation, 88 F.R.D. 211, 218 (D.C.N.D.Ill.1980). See Weit v. Continental Illinois Nat'l Bank & Trust Co., 641 F.2d 457, 469 (7th Cir. 1981); Lupia v. Stella D'Oro Bisquit Co., 586 F.2d 1163, 1168-69 (7th Cir. 1978), cert. denied, 440 U.S. 982, 99 S.Ct. 1791, 60 L.Ed.2d 242 (1979). As the Seventh Circuit Court of Appeals noted in Weit and Lupia, this test is sometimes thought of in terms of standing, and at other times as requiring that direct injury be alleged. The distinction between these two theories is vague at best, and, in this case at least, is of little or no importance. "The fundamental requirement is that plaintiffs establish a sufficient nexus between the defendant's alleged actions and an injury to plaintiffs." Weit, 641 F.2d at 469. The Lupia and Weit cases illustrate how this requirement may be applied. In Lupia, the plaintiff, a food distributor, alleged that the defendant, a wholesaler, violated the antitrust laws by granting certain retail outlets a discount and charging it to the *1390 plaintiff. In Weit, bank charge card holders brought suit alleging that the defendant banks had conspired to fix the interest rates charged for extended payments. In each case the Seventh Circuit held that the plaintiffs failed to establish that they were directly affected by the defendants' anticompetitive acts. In Lupia, only retailers who were not given the discount were within the target area. In Weit, only the charge card customers of a particular defendant bank were directly affected — not the plaintiffs, customers of another bank who sought to represent the other customers through a class action. These cases are analogous to the present one. Perry has alleged that Hartz utilizes illegal tactics against retailers to gain control of the market for its products. Perry's injury, the loss of his job, is not a direct result of Hartz's alleged anticompetitive activities. Only the retailers subjected to Hartz's alleged practices could claim direct injury. Only they are in the target area, and Perry may not sue Hartz as their surrogate. The Court is aware that a panel of the Ninth Circuit Court of Appeals, which was a leader in developing the target area test, see In re Multidistrict Vehicle Air Pollution M.D.L. No. 31, 481 F.2d 122 (9th Cir.), cert. denied, 414 U.S. 1045, 94 S.Ct. 551, 38 L.Ed.2d 336 (1973), has recently forsaken that approach. In Ostrofe v. H.S. Crocker Co., 670 F.2d 1378 (9th Cir. 1982), the panel majority permitted an employee who alleged he was pressured to resign in retaliation for refusing to take part in anticompetitive practices to bring an antitrust suit against his employer. The decision was based almost exclusively on policy grounds, and it is not without its appeal. However, as Judge Kennedy's strong dissent notes, the Ostrofe decision is a striking departure from established precedent in its own, and this, circuit. Without an indication that the Seventh Circuit would join in this departure, this Court will not presume to alter this circuit's settled approach. Perry has not shown that he was in the target area of Hartz's alleged anticompetitive practices. Therefore, his federal antitrust claim is dismissed. Perry also lacks standing under state antitrust law. The Indiana Antitrust Act, particularly I.C. 24-1-2-1 and 24-1-2-2, is patterned after the Sherman Act. Photovest Corp. v. Fotomat Corp., 606 F.2d 704 (7th Cir. 1979), cert. denied, 445 U.S. 917, 100 S.Ct. 1278, 63 L.Ed.2d 601 (1980); Orion's Belt, Inc. v. Kayser-Roth Corp., 433 F.Supp. 301 (D.C.S.D.Ind.1977); Citizens Nat'l Bank v. First Nat'l Bank, 165 Ind. App. 116, 331 N.E.2d 471 (1975). Federal case law has been consulted in interpreting the Indiana statute. Citizens Nat'l Bank, 331 N.E.2d at 478-79. Although the standing issue in Citizens Nat'l Bank is not factually analogous, the court's adherence to the direct-injury requirement, id. at 478-79, supports the notion that federal and state standing tests are essentially the same. Therefore, for the same reasons stated in the discussion of federal antitrust standing, Perry's state antitrust claim is also dismissed. III. Negligent breach of the duty of good faith In Count IV of his complaint, Perry alleges that Hartz's acts constituted a negligent breach of its duty of good faith and fair dealing. Indiana does not recognize that such a duty is owed by an employer to an employee at will. See Campbell v. Eli Lilly Co., Ind.App., 413 N.E.2d 1054, 1066-67 (1980) (Ratliff, J., concurring in part and dissenting in part). For that reason, Count IV is dismissed. IV. Outrageous conduct/intentional infliction of emotional distress Count V alleges that Hartz is guilty of outrageous conduct toward Perry; it is essentially an accusation of intentional infliction of emotional distress. See Restatement (Second) of Torts § 46 (1965). The general rule in Indiana is that claims for emotional distress are not recognized unless the distress is accompanied by a physical injury. Charlie Stuart Oldsmobile, Inc. v. *1391 Smith, 171 Ind.App. 315, 357 N.E.2d 247, 253-54 (1976), on reh. 369 N.E.2d 947 (1977). This rule has one exception: an action for emotional distress unaccompanied by physical injury is permitted when a legal right is invaded in such a way as to "provoke an emotional disturbance." Examples of actions which have been held to fit within this exception are ones for false imprisonment, assault, abduction, wrongful ejection, seduction, and unauthorized autopsy. Id. Although the question of damages for emotional distress has not been considered by an Indiana court in the context of an action charging a retaliatory discharge, this Court believes that the question, if presented, would be resolved in favor of the employee and thus constitute another exception to the general rule. This inference follows from the following language contained in Frampton, supra: We further hold that such a [retaliatory] discharge would constitute an intentional, wrongful act on the part of the employer for which the injured employee is entitled to be fully compensated in damages. To be "fully compensated" is taken by this Court to mean to be compensated for any and all injuries which are proximately caused by the wrongful act. Therefore, if a plaintiff is able to prove by a preponderance of the evidence both that he was the victim of a retaliatory discharge, and that he suffered emotional distress as a result thereof, he should be able to include such element in his measure of damages. However, we fail to perceive that Count V states a tort separate and distinct from that alleged in Count I. The wrongful act allegedly committed by the defendant is to have discharged the plaintiff for an impermissible reason. The emotional distress allegedly suffered is just one of various consequences of that act. For such reason, Count V is surplusage and is dismissed. On trial, the plaintiff will be permitted to attempt to prove emotional distress as a proximate result of wrongful discharge. V. Defamation In Count VI Perry alleges that Hartz defamed him by stating that he was "fired for stealing." There is no indication as to when, where, or to whom the statement was made. As a result, the allegation falls short of stating a claim. Under the prevailing rule in the Seventh Circuit, averments of time and place are material elements of a claim. Kincheloe v. Farmer, 214 F.2d 604 (7th Cir. 1954), cert. denied, 348 U.S. 920, 75 S.Ct. 306, 99 L.Ed. 721 (1955). Observance of this rule helps ensure that a defendant is given a clear idea of the allegations made against him and that the relevance of threshold questions such as those involving statutes of limitations and conflict of laws may be evaluated. Because Perry's defamation claim lacks sufficient allegations as to time and place, it is dismissed with leave to amend. VI. Equitable relief for state antitrust violations Count VII of the complaint requests a decree of ouster under I.C. 24-1-2-5 as a remedy for Hartz's alleged violations of Indiana antitrust law. This count must be dismissed for two reasons: first, as explained in section II of this entry, Perry lacks standing to bring an antitrust suit against Hartz, and second, I.C. 24-1-2-5 authorizes civil suits for violations of the antitrust statute only in the name of the state upon the relation of the proper party, a requirement with which Perry has not complied. Summary Hartz's motion to dismiss is denied as to Count I, but granted as to all other counts of the complaint.

Combined display of video image with superimposed analogue waveforms for clinical applications. An instrument is described which has been developed for use in any clinical investigation where a video image and analogue data are simultaneously acquired and where synchronous viewing of the data is important, both at the time of the study, and at subsequent review. Up to eight analogue signals can be superimposed on the video image, appearing as coloured waveforms scrolling from the right. The output is available in a PAL encoded form suitable for video recording in colour. Resolution is 256 (vertical) by 512 (horizontal) with a maximum sample rate of over 15 kHz. Additional features are: a graticule for calibration, a facility to display alphanumeric characters, and an option for automated uninterrupted viewing of the visual image. The system has been used in urodynamic investigations, for overnight studies of sleep disorders, and for electrophysiological studies during open heart surgery.

Effects of epidermal growth factor, cholecystokinin, and secretin on growth of the alimentary tract in the neonatal guinea pig. Gut hormones and growth factors are likely to be involved in the functional changes and substantial growth of the alimentary tract that occurs during early neonatal life. The present experiments investigated the effects of continuous subcutaneous infusion of cholecystokinin (CCK), secretin, and epidermal growth factor (EGF) in neonatal guinea pigs for 4 or 15 days. At doses of 20, 100 or 500 pmol/kg/h, CCK and secretin had no effect on alimentary tract organs except for an increase in pancreatic weight at 4 days with the highest dose of CCK and an increase in stomach weight at 4 days with the highest dose of secretin. In contrast, EGF showed dose-dependent and time-dependent effects on all the alimentary tract organs when infused at 70, 210 and 630 pmol/kg/h. These results suggest that EGF, but not CCK or secretin, is likely to be an important trophic factor during the neonatal period.

If you are a passenger with special needs, we are here to provide you with the assistance to make your journey as seamless as possible. Lufthansa will not refuse transportation to any individual on the basis of a disability. If you need to speak to someone regarding a special service request regarding your disability, please contact your local Lufthansa office. Pre-travel assistance Prior to your travel, we recommend that you contact us by telephone at 1-800-645-3880 at least 72 hours before your flight to discuss and schedule any additional assistance you may need. TTY (relay) service is available for USA-based hearing impaired customers at 1-866-846-4283 from 8:00 a.m. to midnight EST on business days and from 8:00 a.m. to 10:30 p.m. EST on weekends. Our reservations agents are standing by to assist you with any concerns or questions you may have. Lufthansa Medical Desk If after reading the various topics below, you have a specific medical question that has not been answered, please feel free to contact our Medical Desk. The Lufthansa Medical Desk is open Monday through Friday, from 9:00 a.m. to 3:00 p.m. EST at the following number: 1-516-296-9580. Assistance at the airport and in flight At the airport If you need a wheelchair or other mobility assistance to help you through the airport, we recommend that you contact us 48 hours prior to your flight to schedule additional assistance. If you cannot pre-book, any Lufthansa employee will be more than happy to help you at the airport. In flight assistance We will do our best to assign you to a seat that is most suitable to your needs. Bulkhead seats can be provided if you are traveling with a service animal or if you have an injury that may require a little extra legroom. We also have a number of seats with lifting armrests for ease of access. It is our goal to make your flight as enjoyable as possible. However, pursuant to applicable FAA and safety regulations we might not be able to accommodate your request to be seated at emergency exits, or in cross aisles that form a part of emergency exit rows, if you can not assist in an evacuation of the aircraft. An on board aisle wheelchair is provided on all our flights to assist passengers with transportation to and from the restrooms (please note that crew cannot assist passengers in the restroom). Traveling with your own wheelchair or other assistive device If you are traveling with your own wheelchair, scooter, segway, or other assistive device, we recommend that you contact us with the height and weight information at least 48 hours prior to your journey so that we can make the necessary arrangements. We also request that you come to the airport one hour prior to the recommended check-in time so that we may ensure proper handling of your battery-powered mobility aids. Small collapsible assistive devices that can be safely stowed in an overhead bin or under your seat can be stored by a crew member. Lufthansa will accept assistive devices with batteries as checked baggage as well as inside the cabin. Assistive devices with batteries include respirators, CPAP machines (Continuous Positive Airway Pressure machine), portable oxygen concentrators (POC) and ventilators. Lufthansa will allow qualified individuals with a disability who are using FAA approved personal respirators/ventilators to bring their equipment, including non-spillable batteries, onboard the aircraft. You must have sufficient battery power for 150% of the maximum flight duration. If your wheelchair/assistive device is checked with baggage; it will be returned to you at your final destination. We will transport you in another wheelchair from check in to the aircraft, to any connecting flights, and upon arrival to the baggage collection area at the final destination. If your assistive device is battery powered, please be advised that Lufthansa accepts both spillable and non-spillable batteries. Batteries must be properly labeled and clearly marked “SPILLABLE” or “NON-SPILLABLE” so that our personnel can exercise proper handling procedures for the appropriate battery type. A battery may be refused if there is any sign of damage or leakage. If you cannot pre-book, any Lufthansa employee will be happy to assist you at the airport with coordinating the storage of your assistive devices on the plane. Traveling with an assistant An assistant or companion will be needed to travel with a passenger under the following circumstances: - If the passenger is traveling in a stretcher- If the passenger cannot assist in their own evacuation during an emergency- If the passenger cannot communicate or respond to the crew regarding safety matters- If the passenger cannot comprehend safety instructions due to diminished mental capacity Passengers are reminded that the crew cannot help with breathing, actual eating, the administering of medication, or assist in using lavatory facilities. We are able to help with the identification and opening of food items. We will assist passengers moving to and from the lavatory; yet are unable to assist within the lavatory. Visually impaired passengers If you are visually impaired, we recommend that you contact Lufthansa prior to traveling so that we can make the necessary arrangements. If you cannot pre-book, any Lufthansa employee will be happy to assist you at the airport. We offer the following assistance to our visually impaired passengers:- Separate briefings about safety procedures- Separate briefings about delays and other travel issues- Escort to and from the aircraft- Any other assistance you might need to find locations in the airport Hearing impaired passengers If you are hearing impaired, we recommend that you contact us prior to travelling so that we can make the necessary arrangements. Our TTY number for USA calls is 1-866-846-4283, available Monday-Friday from 08:00 EST – 24:00, Saturday and Sunday from 08:00 EST – 22:30. If you cannot pre-book, any Lufthansa employee will be happy to assist you at the airport. We offer the following assistance to our hearing impaired passengers: - Separate briefings about safety procedures- Separate briefings about delays and other travel issues- Captioning in English & German for the in-flight safety video is available since January 2009 to meet the needs of our hearing impaired passengers Traveling with service/assistance animals If you are bringing a service animal, we recommend that you contact us prior to traveling so that we can make the necessary arrangements. Please be advised that, as a foreign air carrier, Lufthansa is only required to allow dogs on board as a service animal. If you cannot pre-book, any Lufthansa employee will be happy to assist you at the airport. We recommend that all service dogs wear a harness, and that you bring some type of bedding etc. to make the dog more comfortable. On international flights over 8 hours long, we may require documentation showing that your service dog will not need to relieve itself on board, or that the dog can relieve itself in such a way that does not cause sanitation issues or other hazards to your fellow passengers. We recommend that you refer to the government website of the country to which you are traveling for any additional entry requirements or documentation you may need for your dog. Emotional support and psychiatric service animals Lufthansa also welcomes emotional support and psychiatric service dogs. We recommend that you contact us prior to travelling so that we can make the necessary arrangements. If you cannot pre-book, any Lufthansa employee will be happy to assist you at the airport. Required documentation: An emotional support or psychiatric service dog must be accompanied by documentation from a licensed mental health professional stating the following: 1. The passenger has a emotional/mental health-related disability and having the dog on the aircraft is necessary.2. The individual providing the assessment is a licensed mental health professional, and that the passenger is under the care of the individual providing the documentation. 3. The documentation may not be more than one year old. The document must include the date, the type of mental health professional license and the state or other jurisdiction in which it was issued. As our flights are all international flights, we recommend that you refer to the government website of the country to which you are traveling for any additional entry requirements or documentation you may need for your dog. Reporting disability related problems via U.S. DOT Complaint Resolution Official (CRO) If you encounter problems while travelling on Lufthansa, please feel free to ask any crew member or ground staff for a Complaint Resolution Official (CRO). Our CRO’s have been specially trained in sensitivity and awareness, as well as all applicable Federal Aviation Administration (FAA), U.S. Department of Transportation (DOT), Air Carrier Access Act (ACAA) and Americans with Disabilities Act (ADA) regulations and legislation. They will be glad to respond to your concerns. Our CRO’s are available during operating hours at all our U.S. Destinations, and at the following German locations during operating hours: Frankfurt, Munich and Dusseldorf. If you feel that Lufthansa has violated any provision of Title 14, Code of Federal Regulations, Part 382, you may file a formal complaint under the applicable procedures of 14 CFR Part 382.65 via the following channels: U.S. Department of Transportation (DOT) Disability Hotline:If you have experienced time-sensitive, disability related air travel service problems that require immediate attention, you may call 1-800-778-4838 (voice) or 1-800-455-9880 (TTY) to obtain assistance. This hotline is available from 7:00 a.m. until 5:00 p.m. EST, Monday through Friday. Get more information on the U.S. Department of Transportation website under www.dot.gov.

Quench-Shield Ratiometric Upconversion Luminescence Nanoplatform for Biosensing. Upconversion nanoparticles (UCNPs) possess several unique features, but they suffer from surface quenching effects caused by the interaction between the UCNPs and fluorophore. Thus, the use of UCNPs for target-induced emission changes for biosensing and bioimaging has been challenging. In this work, fluorophore and UCNPs are effectively separated by a silica transition layer with a thickness of about 4 nm to diminish the surface quenching effect of the UCNPs, allowing a universal and efficient luminescence resonance energy transfer (LRET) ratiometric upconversion luminescence nanoplatform for biosensing applications. A pH-sensitive fluorescein derivative and Hg(2+)-sensitive rhodamine B were chosen as fluoroionphores to construct the LRET nanoprobes. Both showed satisfactory target-triggered ratiometric upconversion luminescence responses in both solution and live cells, indicating that this strategy may find wide applications in the design of nanoprobes for various biorelated targets.

Mesoscopic membrane physics: concepts, simulations, and selected applications. The window of a few tens to a few hundred nanometers in length scale is a booming field in lipid membrane research, owing largely to two reasons. First, many exciting biophysical and cell biological processes take place within it. Second, experimental techniques manage to zoom in on this sub-optical scale, while computer simulations zoom out to system sizes previously unattainable, and both will be meeting soon. This paper reviews a selection of questions and concepts in this field and demonstrates that they can often be favorably addressed with highly simplified simulation models. Among the topics discussed are membrane adhesion to substrates, mixed lipid bilayers, lipid curvature coupling, pore formation by antimicrobial peptides, composition-driven protein aggregation, and curvature driven vesiculation.

Forkhill Forkhill or Forkill ( , ; ) is a small village and civil parish in south County Armagh, Northern Ireland. It is within the Ring of Gullion and in the 2011 Census it had a recorded population of 498. It lies within the former barony of Orior Upper. Its name, deriving from the Irish word foirceal may refer to the village's position on flat land between the large hills of Tievecrom (to the east) and Croslieve (to the west). History The land in the parish was awarded by Elizabeth I to Capt. Thomas Chatterton and by James II to Lord Audley on condition of English settlement, but by 1659 it was still almost entirely occupied by native Irish people. Following the terms of a trust set up by a subsequent owner, Richard Jackson, much of the property was declared waste and resettled in 1787-91 with a view to encouraging the linen industry, most of the new settlers being Protestants. This was followed by serious breaches of the peace, which have been attributed not to sectarianism but by L. M. Cullen to political disputes among the gentry, and by David Millar to weakening social disciplines on the increasingly independent class of skilled linen weavers, both Protestant and Catholic. The Defenders emerged around Forkhill in the early 1790s, and were responsible in 1791 for an attack on the family of the local schoolmaster, Alexander Berkley, which was so savage as to be remembered with horror and indignation for over hundred years, both in the village and (although the Berkley attack was not provoked by sectarian motives) more widely across the Orange community. Belmont, a military barracks, was set up in the parish in 1795, and a Forkhill Yeomanry in 1796; and a clash with United Irishmen in May 1797 along with the torching of Forkhill Lodge was followed by a policy of military terror which locally undermined the planned 1798 rising to the point of paralysis. 1821 saw the first successful census of Forkhill. About 10% of the 7,063 inhabitants were Protestant; and between the denominations the average family size was identical at just over 5 persons, both employed servants in roughly equal proportions, and both were for the most part situated on small farms. In a national survey in 1835 three local residents responded for Forkhill: Rev. William Smith (Church of Ireland), Rev. Daniel O’Rafferty (Catholic) and Maj. Arthur A. Bernard (Resident Magistrate). They reported that most families were engaged in labouring work, often seeking seasonal jobs elsewhere in the UK, and women no longer worked in the linen trades; rents were £1 a year for a small cabin, or £2 if it had a potato garden. The 1851 census covered language, and found that about a third of the people in the barony spoke Irish (though almost all had at least some knowledge of English). Richard Jackson’s will had provided for free schooling of a Protestant character for the poor children of the parish. In 1825 about a quarter of the parish children were enrolled, of whom about two-thirds were Catholic. Following an intensification of sectarian feeling in the 1820s, Fr Daniel O’Rafferty, the parish priest, applied in 1834 for a national school in the parish, which opened in Maphoner in August. Jackson’s legacy also saw a dispensary set up in Forkhill in 1821, a two-storey building attended by a doctor, Samuel Walker, three days a week and for emergencies. In 1832 violent opposition to the system of tithes reached Forkhill and became associated with Ribbonism, continuing with a series of affrays, attacks and murders until the passage of the Tithe Rent Charge Act 1838 made the tithe payable by landlords. However, this source of resentment soon became overtaken by issues of rent and security, with persistent claims for proper leases being refused in the early 1840s, the Forkhill Estate preferring to deal with tenants-in-chief who sublet on difficult terms, a system giving rise to widespread intimidation and violence without regard to religious affiliation. The famine of the late 1840s, when death and migration reduced the population of the parish by about a quarter, eventually reduced the pressure on land; but before long problems returned, and the nearly-successful assassination of the Killeavy magistrate Meredith Chambré on 20 January 1852 while leaving Forkhill village gave rise to a denunciation by Queen Victoria and the constitution of a Parliamentary Select Committee on Outrages. By the late 1850s pressure by the authorities and by the Catholic Church, the constitution of a petty sessions court in Forkhill, and the Irish Reform Act of 1850 which extended the franchise had all conspired to relieve the situation. On the Troubles of the 1970s-1990s see The Troubles in Forkhill. Development A £550,000 community project comprising retail/office units and a light industrial unit was developed by Forkhill and District Development Association and funded by the Community Regeneration and Improvement Special Programme (CRISP). It opened in May 2004. With the British Army's demilitarization of South Armagh, the site of the old barracks is earmarked for future developments. Sport Forkhill is the home of Forkhill Peadar Ó Doirnín GAC, which is one of the oldest clubs in Armagh GAA, having been founded in 1888. It currently plays Gaelic football at Senior level. Its name commemorates the 18th-century poet Peadar Ó Doirnín. Schools St. Oliver Plunkett's Primary School See also List of civil parishes of County Armagh References Bibliography External links Culture Northern Ireland Category:Villages in County Armagh Category:Civil parishes of County Armagh

Q: How do I convert a .PNG into a .ICO? There's an image in the VS2010 stock icon pack which is only included as a PNG, which I would like to use as an ICO (I want XP and earlier users to see the icon). How can I convert the PNG into the ICO? A: You can use the opensource imagemagick convert utility, that can take several images and pack them in one single icon file. It can also do re-sizing and lots of image manipulation. Imagemagick contains other tools for image handling, is available on multiple platforms (Linux, Windows, Mac Os X, iOS) and can be batched easily, for example in Continuous Integration pipelines. Here is a basic example exporting a svg file to a png file: convert icon.svg -scale 32 tmp/32.png And how to pack several several of such png files in a final ico file: convert tmp/16.png tmp/32.png tmp/48.png tmp/128.png tmp/256.png icon.ico Note that the standalone convert binary may not installed in the default installation. On Windows, you must check the (quite explicit) Install legacy utilities (e.g. convert) option. In case you missed it, simply use the slightly longer magick convert command rather than convert alone. A: For one-off tasks I usually just cheat: ConvertICO.com . If you will be doing this fairly often, you may want to consider the free Photoshop plugin. A: Install ffmpeg from: Windows: http://ffmpeg.zeranoe.com/builds/ OSX: http://ffmpegmac.net/ From the shell, use the following command to convert PNG to ICO. ffmpeg -i img.png img.ico Also if you use ffmpeg regularly, don't forget to add it your PATH variable.

[Notch signaling in bone formation and related skeletal diseases]. Notch signaling is highly conserved in evolution and regarded as a key factor in cell fate determination. It mediates cell-to-cell interactions that are critical for embryonic development and tissue renewal, and is involved in the occurrence and metastasis of neoplasm. Recent researches have found that such signaling plays an important role in modulating the differentiation of chondrocytes, osteoblasts and osteoclasts. Dysfunction of Notch signaling can result in many skeletal diseases such as bone tumor, disorders of bone development or bone metabolism.

I shot these videos this year at the annual Origins game expo in Columbus Ohio, while running the tournament for Looney Labs' card game, Nanofictionary. I actually uploaded them to YouTube within a month this time!.

Candy The parting moments were unexplainable to both of them. They were sipping coffee, gazing around, and occasionally talking. It was a cold dark night of misty December. He was to catch a train scheduled for departure to the front at midnight. There would be many like him sitting in the train leaving behind life that was no more to be their fortune. His mind was divided between her thoughts and those who would soon join him. She wanted to tell him that life would be barren without him, and she would be continuously waiting for him each passing day and night. He too could not tell her that parting had made his soul empty. They both could not say to each other that love was making them weak and, their souls were praying for each other’s safety and happiness. He had always appeared a brave man to her. He was composed, occasionally deeply quiet and often laughing loudly. Tenderness had been successfully hidden in the deep valleys of two hearts. The demeanour of a soldier made him rock-solid in appearance, both in life and in death. She asked him to take the food regularly as he was always casual in eating. He advised her to re-join the university and complete her studies. They both laughed, briefly embraced each other and left the place. A nearby lone tree uttered few sighs, called rustles of the wind, and shed tears of love; mostly given the name, dewdrops. Soon he was among his comrades on the train and found everyone laughing. In a moment he understood the cause behind their laughter, hurriedly hid the swallowing sadness, and laughed loudly, too. They all joined his loud laughter. Suddenly there was a loud thunder outside and rain started pouring down heavily. The train blew the whistle and started moving in the night towards a destination far off. The soldiers’ loud laughter could still be heard! The tears of rain were still pouring down and absorbed by the thirsty womb of the land. …………………………………………………… It was first day of the new year and an air of festivity could be found all around. Actually it was a HQ premises surrounded by high walls and barbed wires. It was eastern part of the city of a war-torn country that had been secured by the soldiers of peace from many nations. With the advancing human civilization, rehabilitation of the citizens had become part of the war norms, though humans still remained unable to get free from shackles of conflict and misery. He was part of the soldiers who had come from distant lands to enforce peace there. She was working in an aid agency and living in the same premises. The HQ had arranged a marathon race as part of new year festivities. The participants were to cover the distance in shape of many rounds of the vast compound building. He was standing along the roadside amidst the cheering crowd. She had passed him few rounds and every time he yelled encouraging words to her. The moment she finished the race, he came forward with a bottle of water, and also gave her a candy. “Sorry, I had only just one left,” he said sheepishly. She gave him a bright smile and moved towards the victory stand. It did not take them long to find out that they both were from the same land, though her family had left the homeland many decades ago and had settled in a new country. The singularity of Motherland became the cause of frequent union of the two in the following months. One day she told him that she had not thrown away the wrapper of the candy he gave it to her at the end of the race, and preserved it in her notebook. He was amused to the soul, and held her hand firmly. They both giggled and started walking on the long empty road. …………………………………………………… He had gone to the front again. This time the war had engulfed his own homeland. It was such a devastating simmering war that it had lasted for more than a decade. She was passing days and nights in company of hope and courage. He should have been back by then after completing his tour of duty, but the intensity of the war had further thickened, and absorbed many young mortals. The life could have passed on this pattern had it not been revealed to her by the doctor that her life had come to a sudden end. The long bouts of coughing for the past few months finally compelled her to go to the doctor. “The disease has prevailed over a period of time and the tumour has spread to many parts, cure is almost impossible,” the doctor said briefly in a flat tone. All seemed lost to the destiny including the unconquerable will to live. One night she suffered a severe bout of cough mixed with blood. She did not take long to understand that parting moments had come and were not far away. That night, she mustered the left over courage, and started writing a letter to him. She picked her notebook and selected a page to write. The letter to a soldier who had been a marvellous companion but was part of a war that seemed to have no end. She wanted to write everything: her broken dreams and lost hopes, her darkened future and shattered life. But she could not write except how much she loved him and had prayed to spend the whole life with him. She could hardly scribble few more lines that a fresh bout of severe blood coughing overpowered her. She hurriedly tore the page and sealed it in an envelope. Inside the envelop, she did not forget to put the candy wrapper! …………………………………………………… On that dark night when love was parting, a companion was dying, a soldier was fighting, the gales of wind sang through the trees uniting the two souls; whispered Lord Byron: When we two parted In silence and tears, Half broken-hearted To sever for years, Pale grew thy cheek and cold, Colder thy kiss; Truly that hour foretold Sorrow to this. The dew of the morning Sunk chill on my brow – It felt like the warning Of what I feel now. Thy vows are all broken, And light is thy fame; I hear thy name spoken, And share in its shame. They name thee before me, A knell to mine ear; A shudder comes o’er me – Why wert thou so dear? They know not I knew thee, Who knew thee too well – Long, long shall I rue thee, Too deeply to tell. In secret we met – In silence I grieve, That thy heart could forget, Thy spirit deceive. If I should meet thee After long years, How should I greet thee? – With silence and tears. …………………………………………………… Far away from life, in a field hospital, a soldier’s leg was amputated as the only option to save his life! …………………………………………………… There sat four friends in a field under the open sky full of stars. They were: a poet, a teacher, a farmer, and a soldier. They lit heavy fire and sat around as it was a cold December night. It was their routine to sit and talk for the whole night and depart at the break of dawn. They were to define four different lives on the coming day with a new hope for the land. Mostly their talk was abrupt and followed no pattern. However, every man stood his ground and shared what he had lived so far and wished to do in coming years. The poet always narrated tales of a new land that would not bear any traces of human sufferings. He desisted any signs of injustice, inequality and exploitation prevailing on the land. The teacher would often tell them about the power of a “construct” that how minds could be shaped using particular “frame of references”. He often unveiled the mysteries of cognition and behaviour. To him, a new life was possible by shaping the phenomenon which is called “education”. The farmer usually orchestrated his talk around types of land and crops. He was of the opinion that every land has a power to yield new crops but it is up to farmers to identify and sow the proper seeds. Having done this, the land would never disappoint. The soldier was plainest in his talk. He would remember his comrades who lost their lives and limbs in the battlefield. His stories were painful but his voice was full of hope and resolve. He could eulogise the Motherland for the whole night without a glimpse of tiredness or lethargy. While talking and giving them lessons “to fight till end”, he would never look at his amputated leg! …………………………………………………… All four joined in unison, “Friends! Rise to construct your own world. Do not only point fingers and complain. Have the will to lose, to find new meanings of life. Do not wait for the brave ones to deliver for you, be the doer of deeds, and keep doing! Acquire the wisdom to pursue achievable goals, cultivate dreams but not the ghosts of velvet wishes. Rise, and be defined by act, and be not the merchants of hollow words. Work, produce, and distribute! There is always ‘new’ to construct. It is a tough task or calling but men of self-worth do embark on journeys to hitherto unfound lands.” It was break of the dawn. The new sun was on the horizon. They all shook hands and together left for a new destination. A gradually diminishing noise of the crutches of the man with amputated leg could be heard for long. The shadow of ‘work’ was following all four! …………………………………………………… The journey of life would continue. There would be a girl, a soldier, a candy wrapper, and a story. The womb of Motherland is too vast and warm to absorb the sufferings of all. Its only claim is ‘unconditional love’. (The writer is a traveller and student of human history. He can be reached at tmabbasi@yahoo.com)

A comparative analysis of tissue expander reconstruction of burned and unburned chest and breasts using endoscopic and open techniques. Tissue expansion is not widely accepted for reconstruction of breast and chest burn deformities because of concerns about the capacity of compromised skin to stretch without complications. The authors hypothesized that tissue expander reconstruction of breast and chest burn deformities is reliable and has outcomes similar to those of expansion of similar nonburned tissues. The authors used congenital breast anomalies as a control because they share similar reconstructive challenges: constricted skin envelope and gross malformation of the parenchyma and nipple-areola complex. The authors also hypothesized that endoscopic techniques may improve outcomes for breast and chest burn reconstruction. A retrospective review was completed of tissue expander reconstructions of burn and congenital breast deformities. All reconstructions used an endoscopic or open tissue expander placement and subsequent local tissue rearrangements. Data were analyzed using parametric and nonparametric methods. For reconstruction of burn deformities, 15 women had 37 expanders placed. Within the congenital breast cohort, 20 patients had 22 tissue expanders placed. There were no statistical differences in follow-up time, body mass index, or comorbidities between burn and congenital patients. There was no statistical difference in major complications (p = 0.72) between these groups. Within the burn deformity cohort, endoscopic reconstructions had fewer major complications (p = 0.04), required less operative time per expander (p < 0.001), and required less time to expand (p = 0.021). The authors believe that breast and chest burn deformities can be safely reconstructed with tissue expanders without increased complications over expander reconstruction of the congenital breast. Furthermore, endoscopic techniques may be superior for burn deformities because of improved visualization and remote incisions.

Kotateou Kotateou is a village in the Bassar Prefecture in the Kara Region of north-western Togo. References External links Satellite map at Maplandia.com Category:Populated places in Kara Region Category:Bassar Prefecture

Q: Python Compare and Update CSV File I have two CSV files. csv_1 columns: AP Name Mac Switch Port AP-2-2 001122334455 switchname1 0/37 AP-2-3 554433221100 switchname2 0/41 csv_2 columns: Mac Switch Port 001122334455 switchname1 0/37 554433221100 switchname2 0/41 I want to update the switch and port columns in csv_1 with the switch and port in csv_2 based on when a mac address match found (these are not in order). What is the best and most efficient way to do this properly in python? I know how to read in the CSV files, I'm just not sure how to check the values properly. A: You can use pandas to join in the new values: import pandas as pd df1 = pd.read_csv("<path to csv1>.csv") df2 = pd.read_csv("<path to csv2>.csv").rename(columns={"Switch": "new_switch", "Port": "new_port"}) # clean up column names df1.columns = [i.strip() for i in df1.columns] df2.columns = [i.strip() for i in df2.columns] # join in new values result = df1.join(df2.set_index("Mac"), on="Mac") # use the new values where they're defined, otherwise fill in with the old values result["Switch"] = result["new_switch"].fillna(result["Switch"]) result["Port"] = result["new_port"].fillna(result["Port"]) # delete the "new" columns, which are no longer needed del result["new_switch"], result["new_port"] # write out result.to_csv("<path to new csv>.csv", index=False) This assumes your csv2 has only one row for each Mac value.

Q: Manipulating expressions in R I am looking for a way to create an expression that is the product of two given expressions. For example, suppose I have e1 <- expression(a+b*x) e2 <- expression(c+d*x) Now I want to create programatically the expression (e1)*(e2): expression((a+b*x)*(c+d*x)) Background I am writing a model fitting function. The model has two pieces that are user-defined. I need to be able to "handle" them separately, and then create a combined expression and "handle" it as one model. "Handling" involves taking numeric derivatives, and the deriv function wants expressions as an input. A: Try this: e1 <- quote(a+b*x) # or expression(*)[[1]] e2 <- quote(c+d*x) substitute(e1 * e2, list(e1=e1, e2=e2)) A: I don't deal with this too often but something like this seems to be working e1 <- expression(a + b*x) e2 <- expression(c + d*x) substitute(expression(e1*e2), list(e1 = e1[[1]], e2 = e2[[1]])) # expression((a + b * x) * (c + d * x))

Financial Services Finding you the best mortgage to suit your needs. McRobieAdams are well established Mortgage and Protection Advisers with a Head Office in Banbury, a branch in Brackley and staff in Witney and Abingdon. Providing a professional service and priding ourselves on looking after our clients old and new. We have access to the whole of the market enabling us to find the best mortgage to suit a client’s needs. Extensive business to business relationships help us provide a leading service to our corporate and personal clients. Directors Kevin McRobie With 30 years’ experience in Financial Services including banking, building society and business development experience. This extensive background helps provide a comprehensive service to everyone including local industry related business clients. McRobieAdams has grown significantly in recent years and has been invited to help provide services for a large estate agency business. Contact 01295 271888 Adam Messer Started Financial Services over 9 years ago with a large national estate agency chain and progressed to managing a team of advisers. The breadth of experience allows Adam to fully understand what is needed by the estate agents we work closely with and how important it is to keep everyone informed in order to keep the buying chain together. Contact 01295 271888 Mortgage & Protection Advisers Rachel Smith Rachel first started in the Financial Services Industry over 20 years ago, spending most of her time with a large corporate estate agency chain as a Mortgage Advisor. More recently she has been successfully managing the Mortgage Administration Department at McRobieAdams. Her experience allows Rachel to fully understand what is needed by all parties involved in the mortgage process and she prides herself on her customer service. Contact 01295 271888 Nigel Riley Nigel has worked in the Financial Services industry since 1994 with experience in both corporate and retail sides of the business. He joined McrobieAdams at the beginning of 2016 to help grow the company further and has an in depth and wide ranging knowledge of all aspects of the mortgage market. He uses this knowledge and experience to provide the highest level of service to every client. Contact Brackley 01280 843000 Emma Horwood Having worked in the financial industry for over 17 years and as a qualified Mortgage adviser for over 11 of these, Emma prides herself on her friendly and knowledgeable approach to this sometimes daunting process. She has the added advantage of working in the Estate Agency Field managing a successful branch for two and a half years so is able to offer helpful advice through any stage of your house buying process. Contact details Witney 01993 773730/ 07842 694430 Selina Targett Selina has been a qualified mortgage adviser since 2007 and has previously worked for large corporate estate agencies. An excellent working relationship with a number of local estate agents in Abingdon and the surrounding area. Selina uses her experience to guide clients through the house buying process and offers a personal and friendly approach helping clients find the best mortgage suited to their individual requirements. Selina prides herself on the way that she looks after her clients.

<!doctype html> <html> <head> <meta charset="utf-8"> <title>{{ "MailTitle" | get_plugin_lang('AdvancedSubscriptionPlugin') | format(session.name) }}</title> </head> <body> <table width="700" border="0" cellspacing="0" cellpadding="0"> <tr> <td><img src="{{ _p.web_plugin }}advanced_subscription/views/img/header.png" width="700" height="20" alt=""></td> </tr> <tr> <td><img src="{{ _p.web_plugin }}advanced_subscription/views/img/line.png" width="700" height="25" alt=""></td> </tr> <tr> <td valign="top"><table width="700" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="50">&nbsp;</td> <td width="394"><img src="{{ _p.web_css }}/themes/{{ "stylesheets"|api_get_setting }}/images/header-logo.png" width="230" height="60" alt="Ministerio de Educación"></td> <td width="50">&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td>&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td>&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td style="color: #93c5cd; font-family: Times New Roman, Times, serif; font-size: 24px; font-weight: bold; border-bottom-width: 2px; border-bottom-style: solid; border-bottom-color: #93c5cd;">{{ "MailTitleAdminAcceptToStudent" | get_plugin_lang('AdvancedSubscriptionPlugin') | format(session.name) }}</td> <td>&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td>&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td>&nbsp;</td> </tr> <tr> <td height="356">&nbsp;</td> <td valign="top"><p> {{ "MailDear" | get_plugin_lang('AdvancedSubscriptionPlugin') }} </p> <h2>{{ student.complete_name }}</h2> <p>{{ "MailContentAdminAcceptToStudent" | get_plugin_lang('AdvancedSubscriptionPlugin') | format(session.name, session.date_start) }}</p> <p>{{ "MailThankYou" | get_plugin_lang('AdvancedSubscriptionPlugin') }}</p> <h3>{{ signature }}</h3></td> <td>&nbsp;</td> </tr> <tr> <td width="50">&nbsp;</td> <td>&nbsp;</td> <td width="50">&nbsp;</td> </tr> </table></td> </tr> <tr> <td><img src="{{ _p.web_plugin }}advanced_subscription/views/img/line.png" width="700" height="25" alt=""></td> </tr> <tr> <td><img src="{{ _p.web_plugin }}advanced_subscription/views/img/footer.png" width="700" height="20" alt=""></td> </tr> <tr> <td>&nbsp;</td> </tr> </table> </body> </html>

Progressive amyotrophy as a late complication of myelopathy. A delayed syndrome of progressive weakness has been described in survivors of paralytic poliomyelitis - "Post-Polio Muscular Atrophy (PPMA)". One proposed etiology is a drop-out of motor neurons due to increased metabolic demands of an enlarged motor unit territory. We report a patient with slowly progressive lower extremity weakness 20 years after recovery from an episode of myelopathy involving the lower lumbar and sacral segments of the spinal cord. Delayed progressive amyotrophy may complicate any significant injury to anterior horn cells.

Workspace Email Help Use spam filter settings While junk email might be worth a laugh or two, it's not funny when it overruns your Inbox. You can use Workspace Webmail's features for spam management to prevent unsolicited junk email from clogging up your Inbox. Blocking email addresses (example@coolexample.com) or domain names (coolexample.com) treats emails from those email addresses or domains as spam, and then filters them based on your spam settings. Allowing email addresses or domain names never marks email from them as spam. If you select Restricted Mode when you set up spam filters, senders must be on your Allowed List for their emails to deliver to your Inbox. To Set Up Allowed and Blocked Senders Custom Filters — Message filtering allows you to send specific messages to appropriate folders that you have set up. Allowed List — Allowing email addresses (example@coolexample.com) or a domain name (coolexample.com) never marks email from them as spam. Blocked List — Blocking email addresses (example@coolexample.com) or a domain name (coolexample.com) treats emails from those email addresses or domains as spam, and filters them based on your spam settings. Auto Purge List — Purging email addresses (example@coolexample.com) or domain names (coolexample.com) immediately purges all emails from those email addresses or domains, so you never see them. Click Add New, and then OK when you are finished. To remove email addresses or domains from the Allowed, Blocked, or Auto Purge list tab, select it, and then click Delete. Was This Article Helpful? Thanks for your feedback. To speak with a customer service representative, please use the support phone number or chat option above. Glad we helped! Anything more we can do for you? Sorry about that. Tell us what was confusing or why the solution didn’t solve your problem.

Every time you cheat in G+, a bunny commits suicide. Stop the torture. My own simple rules to save the bunnies:● don't use artwork or photos without giving credit to the owner.● don't copy/paste content from the web without a link to the source.● don't post other people's content or ideas as your own ➜ goo.gl/eWBmQ What I don't understand, though, is why bunnies should be psychologically vulnerable to plagiarism on G+ (unless they are the original authors). It might make more sense to say "a bunny gets it", but that would seem to be incredibly harsh on our cute, furry, myxomatosis ridden little friends From the agency websiteDomestic and International copyright and trademark laws protect the entire contents of the Site. The owners of the intellectual property, copyrights and trademarks are WEBM, its affiliates and subsidiaries or other third party licensors. YOU MAY NOT MODIFY, COPY, REPRODUCE, REPUBLISH, UPLOAD, POST, TRANSMIT, OR DISTRIBUTE, IN ANY MANNER, THE MATERIAL ON THE SITE, INCLUDING TEXT, GRAPHICS, CODE AND/OR SOFTWARE. giving a credit when posting an image from another source does not mean you can breech the copyright of the author without the express written permission of said author I have an even better one. Try coming up with original content. Too many, including Adelstein see a popular post under Explore, then go and simply copy it instead of resharing it to make it look like they are publishing something new. I am neither an artist nor photographer, so I don't speak about my own content. What is so difficult to mention the sources for a good photo or artwork in a post +Joe Mama, +Chuck Green, +Johann Blake? +Ali Adelstein It's so difficult cause people suck. Most of my profile is shared content, with links and credit given. Those that aren't, are mine. As for my phone backgrounds and pic messages to friends, I don't care who made it and neither do they. In my opinion, any artist with real talent, shouldn't have a problem getting their portfolios published. Published means copyright protection, and it's much harder to copy and paste out of a book. Hmmmz. +Ali Adelstein why would it not be enough to just post a link for sharing. The source is the Internet and any info on that site. Now, just a photo or image I can see needing to be more attentive to sourcing details. The third bullet is just wrong. I have noticed that some people don't use quotation marks. If I pull others work, I will just put it in quotes. 

Q: Immutable.js and flatMap equivalent function Was just wondering if there an equivalent flatMap function in Immutable.js? I have been using this node package https://www.npmjs.com/package/flatmap for a while but I would prefer to write things like this listObject.flatMap(x => ...) Instead of flatMap(listObject, x => ...) A: Actually, there is a flatMap function (see documentation). However, if you ever face a similar problem (you want to use foo.bar(args) instead of bar(foo, args)) you can create a custom property of your instance. listObject.flatMap = a => flatMap(listObject, a); And after that it is equivalent to write flatMap(listObject, x => ...) and listObject.flatMap(x => ...).

Introduction {#sec1} ============ Scleromalacia or necrotizing scleritis may occur after pterygium excision, trauma, infectious scleritis, or surgery for retinal detachment repair and may also occur in patients with systemic autoimmune diseases such as rheumatoid arthritis, Wegener\'s granulomatosis, and collagen vascular disease.[@bib1] To reinforce thinned sclera, various materials have been attempted for grafting including allogeneic sclera, amniotic membrane (AM), fascia lata, dura mater, pericardium and periosteum.[@bib2]^--^[@bib6] However, an ectopic donor tissue such as periosteum, dura mater, or pericardium is hard to obtain; therefore, clinical experience with these has been limited. Among these materials, preserved donor scleral grafts have been preferred for a long time and have produced acceptable outcomes.[@bib7]^--^[@bib11] Because epithelial wound healing is delayed on a scleral graft, a permanent AM transplantation or a conjunctival graft is usually combined with scleral grafting, which consumes lots of surgical time.[@bib9] Currently, both human preserved donor corneas and scleras are obtainable from a number of eye banks and are readily available for surgery. Corneas and scleras are histologically similar in that they are organized mostly from type I collagen fibers.[@bib12] However, compared with the sclera, corneal collagen fibers are highly uniform in diameter, and they are regularly and compactly arranged. Moreover, cornea has a subepithelial basement membrane consisting of collagen type IV, laminin, and fibronectin that may affect wound healing, whereas the external surface of the sclera is bound to the overlying episcleral only by very thin bands of collagen.[@bib13]^,^[@bib14] Several reports have stressed the favorable effects of scleral grafts, which were well-organized in the perforated corneal area corneal perforated area; however, it is worrying that the collagen fibrils of the scleral graft were broken down by fraying into microfibrils, then shrank and became rapidly cleared.[@bib15]^--^[@bib17] Our hypothesis is that a composition of basement membrane and compact and regular collagen fibrils of the preserved cornea may promote wound healing and may prolong survival as a graft in scleromalacia. Therefore, in the present study, we produced a scleral thinning model by lamellar dissection of the scleras in rabbit eyes. Using this model, we then investigated the effect of preserved corneal grafting on wound healing and inflammation and compared the effects of corneal grafting with those of preserved scleral grafting. Methods {#sec2} ======= Study Design {#sec2-1} ------------ Our study design was outlined as follows:1.Prepare scleral defect rabbit models.2.Transplant preserved human corneas and preserved human scleras.3.Observe and evaluate the wound healing rate, vascularization, and filament formation over 3 weeks.4.Histologically evaluate the inflammatory cells or pro-angiogenic stem cells in the engrafted donor--recipient complex. Animals and Animal Models {#sec2-2} ------------------------- Animal experiments were performed in accordance with the ARVO Statement for Use of Animals in Ophthalmic Vision and Research, and the protocols were approved by the Institutional Animal Care and Use Committee of Seoul National University Biomedical Research Institute (IACUC No. 18-0138-S1A0). Eight-week-old female New Zealand White rabbits (total 10 eyes) weighing 2.0 to 3.0 kg (Orient Bio, Inc., Seongnam-si, Gyeonggi-do, Korea) were used in this study. Five eyes were assigned to each of two groups: the corneal grafting group and the scleral grafting group. A combination of tiletamine and zolazepam-mixed agent (10 mg/kg; Zoletil, Virbac, Fort Worth, TX, USA) and xylazine (2 mg/kg; Rompun, Bayer, Leverkusen, Germany) was injected intramuscularly for anesthesia. Corneolimbal bridle suture was done using 8-0 Vicryl suture to expose the superotemporal sclera for lamellar dissection. Both the conjunctiva and sclera were trephined using a 6.0-mm Barron vacuum trephine (Katena Products, Denville, NJ) with approximately 250 µm of partial scleral thickness at the superotemporal area, 2 mm apart from the corneal limbus and temporal margin of superior rectus muscle. Thereafter, roundly cut scleral tissue was subjected to lamellar dissection using a \#69 Beaver Mini-Blade (Beaver-Visitec, Waltham, MA) as a scleral defect rabbit model. Preserved Human Donor Cornea and Scleras {#sec2-3} ---------------------------------------- Preserved human donor full-thickness corneas contained in glycerol were obtained from Eversight Eye Bank (Cleveland, OH). Preserved human donor scleras contained in 95% ethanol were obtained from Central Ohio Lions Eye Bank (Columbus, OH). The usage of human donor tissues was approved by the Institutional Review Board of Seoul National University Biomedical Research Institute (IRB No. E-1808-028-963; Jongno-gu, Seoul, Korea). Transplantation of the Graft {#sec2-4} ---------------------------- Representative photos of serial surgical procedures are shown in [Figure 1](#fig1){ref-type="fig"}. The preserved donor corneas and scleras were washed and soaked in BSS sterile irrigating solution (Alcon, Fort Worth, TX), then Descemet\'s membrane was peeled off from each cornea and remnant uveal tissues were completely removed from each sclera before use. For grafting, corneal and scleral tissues were trephined into 6.0-mm diameter pieces using a 6.0-mm Barron vacuum trephine (Katena Products). Prepared corneal grafts with the basement membranes facing up or scleral graft tissues were placed on the scleral defect areas, and were then fixated by eight simple interrupted sutures using 10-0 nylon. Levofloxacin eye drops (Cravit; Santen Pharmaceutical Co., Ltd., Osaka, Japan) were instilled to prevent infection at the end of the procedure. {#fig1} Observations and Outcome Measurements {#sec2-5} ------------------------------------- After surgery, we performed time--serial observations of the surgical wounds for 3 weeks, especially on postoperative days 3, 5, 7, 9, 14, and 21. When a sutured area became epithelized, the relevant sutures were removed over the follow-up period. Every observation was done under anesthesia by the intramuscular injection of a combination of tiletamine and zolazepam-mixed agent and xylazine. Surgical wounds were photodocumented in all eyes with and without fluorescein dye staining using a digital camera under a surgical microscope. At postoperative day 21, all rabbits were sacrificed, and the circular cornea- or sclera-engrafted areas were excised 2 mm wider than the circular junction of the donor tissue and the recipient conjunctivas and scleras. We established four outcome measurements, including the wound healing index, surface filament formation, vascularization (perigraft and intragraft), and histologic evaluation of the whole graft, to evaluate the postsurgical wound healing and inflammation at the area of engraftment. Wound Healing Index {#sec2-6} ------------------- The epithelization rate was analyzed to evaluate wound healing over the surface at the engrafted area. The total graft area and deepithelized area (i.e., fluorescein dye-stained area) were estimated based on the acquired photos using the ImageJ software ver. 1.46 (National Institutes of Health, Bethesda, MD; <http://rsbweb.nih.gov/ij/>). The extent of wound healing was determined by the proportional area of epithelization relative to the total graft area and was presented as a wound healing index (%) equal to the epithelized wound area divided by the total graft surface area. Estimations of Perigraft and Intragraft Vascularization {#sec2-7} ------------------------------------------------------- We quantified the densities of intragraft and perigraft vasculature using ImageJ software to investigate the vascularization over the graft and the vascular hyperemia near the graft, respectively. First, the photos used for estimation were opened in ImageJ and were changed into eight-bit images. Then, we measured the estimated area of vascularization by thresholding the vasculature with a cutoff value of 160 for perigraft analysis and with a cutoff value of 180 for intragraft analysis. For perigraft analysis, we drew an imaginary ring of 1.5-mm thickness around the circular graft in a photo. Then, only the quarter-ring from 2 to 4 o\'clock adjacent to the corneal limbus was included in the perigraft analysis. We excluded the vasculature within the opposite half of the ring because it includes the vessels from the extraocular muscle and conjunctiva. For intragraft analysis, the vessel density was estimated only from the inner circle with a 4-mm diameter, not including the peripheral vessels at the graft, to determine whether there were vascularized vessels over the graft surface or perigraft hyperemia. Histologic Evaluation {#sec2-8} --------------------- The excised transplanted graft-recipient bed complexes were fixed in 10% formaldehyde and embedded in paraffin. The formalin-fixed graft was cut into 4-µm thicknesses and placed on microscope slides. After deparaffinization, tissue sections were stained with hematoxylin or incubated with rat monoclonal antibody against CD3 (1:100; \#14017.7, Abcam, Cambridge, MA) and rabbit polyclonal CD34 (1:100; \#ABIN676898, Antibodies-online GmbH, Aachen, Germany) overnight at 4°C, and were then incubated with secondary antibodies conjugated with rhodamine (1:500; \#AP136R, Merck Millipore, Burlington, MA) and Alexa Fluor 488 (1:500; \#A11034, Invitrogen), respectively, for 1 hour at room temperature. After washing with phosphate buffered solution, cover slips were mounted using fluorescent mounting medium with DAPI (\#E19-18, GBI Labs, Bothell, WA). The sections were examined under a microscope (BX53; Olympus, Tokyo, Japan) and photodocumented. To count inflammatory cells and CD3^+^ or CD34^+^ cells, respectively, we selected five square spots (0.1 mm^2^ area per spot) in the section of hematoxylin and eosin stain for inflammatory cells and three square spots (0.1 mm^2^ area per spot) in the section of immune stain for CD3 and CD34. Then, the average numbers of the relevant cells were calculated in each eye. Statistical Analysis {#sec2-9} -------------------- The data are presented as the mean ± standard error measurement. Statistical analyses were performed using the SPSS software version 20.0 (SPSS, Inc., Chicago, IL) and GraphPad Prism v.8.1.2 (GraphPad Software, La Jolla, CA). Differences between two values were analyzed by Mann--Whitney *U* test, and categorical data were analyzed using McNemar\'s test. A *P* value of less than 0.05 was considered statistically significant. Results {#sec3} ======= Epithelial Wound Healing {#sec3-1} ------------------------ The surface of the grafts became nearly full epithelized by 21 days in both groups, although the wound healing index in the corneal grafting group was constantly higher throughout the follow-up period, with significance especially at 9 days after surgery (*P* = 0.032, Mann--Whitney *U* test; [Fig. 2](#fig2){ref-type="fig"}). The standard deviation of the wound healing indices between objects was markedly lower in the corneal grafting group after 7 days. There was no filament formation in the surfaces of corneal grafts during any of the follow-up period. Although without statistical difference between the two groups, four eyes (80 %) of the total five eyes with scleral grafts had copious filaments over the graft surface at day 14 ([Fig. 3](#fig3){ref-type="fig"} and [Supplementary Table S1](#tvst-9-7-38_s002){ref-type="supplementary-material"}). {#fig2} {#fig3} Vascularization of Grafts {#sec3-2} ------------------------- At the end of the follow-up period, the mean area of perigraft vascularization adjacent to the corneal limbus and the vascularized area on the graft surface tended to be higher in the scleral grafting group ([Fig. 4](#fig4){ref-type="fig"}), although the difference was not significant. However, in some of the cases with scleral grafts, the perigraft vessels seemed highly dense, edematous and engorged extending to the margins of the grafts. {#fig4} Histopathology of Grafts {#sec3-3} ------------------------ In sections, engrafted donor corneas still nearly held their original shapes, while the graft junctions of the engrafted scleras became inconspicuous ([Fig. 5](#fig5){ref-type="fig"}A). The average number of inflammatory cells at the graft areas was significantly higher in the scleral grafting group (63.7 ± 16.1 cells/0.01 mm^2^) than in the corneal lamellar grafting group (6.8 ± 4.1 cells/0.01 mm^2^; *P* = 0.016, Mann--Whitney *U* test; [Fig. 5](#fig5){ref-type="fig"}B). Although the inflammatory cells were mostly confined to the graft--recipient junction in the corneal grafting group, the inflammatory cells were found not only at the graft junction, but also throughout the entire graft in the scleral grafting groups. The outermost layers of the engrafted corneas were composed of three to four cell layers of stratified squamous epithelial cells with constant thickness, whereas copious inflammatory cells were observed throughout the surface epithelial layers in the engrafted scleras to disrupt the intact epithelial layers, resulting in uneven outermost surfaces ([Fig. 5](#fig5){ref-type="fig"}C). The stromal layers were loosened in the scleral grafts, and their bitemporal margins were replaced with inflammatory cells and appeared melted. In the engrafted corneas, compact collagen layers were well-preserved and junctional structures were well-retained. Unlike the corneal grafts, there were a number of inflammatory cells at the graft--recipient junctions in the scleral grafts associated with the marginal graft melting. {#fig5} Next, we investigated the phenotypical features of CD3**^+^** T cells and CD34**^+^**endothelial progenitor cells in the inflammatory area at the graft-recipient junctions in both groups. CD3**^+^** T cells and CD34**^+^** cells were identified at the margins of the grafts, and they were colocalized mostly in the same area ([Figs. 6](#fig6){ref-type="fig"}A and [6](#fig6){ref-type="fig"}B), which suggests that perigraft vascularization may be a secondary phenomenon accompanied by inflammatory reaction. The mean number of CD3^+^ T cells tended to be slightly higher in the scleral grafting group (55.9 ± 7.0 cells/0.01 mm^2^) than in the corneal lamellar grafting group (47.4 ± 7.8 cells/0.01 mm^2^; *P* = 0.841, Mann--Whitney *U* test; [Fig. 6](#fig6){ref-type="fig"}C), although it was not significantly different. There was no difference in the numbers of CD34^+^ cells between the two groups (36.1 ± 6.0 cells/0.01 mm^2^ in the corneal grafting group and 36.1 ± 5.3 cells/0.01 mm^2^ in the scleral grafting group; *P* = 0.841, Mann--Whitney *U* test). {#fig6} Discussion {#sec4} ========== In this study, we newly applied the preserved human cornea in lamellar grafting in a scleral defect rabbit model and evaluated the surgical usefulness of the preserved cornea compared with the preserved sclera. We found that wound healing was more rapid, less inflammatory, and caused less vascular engorgement in corneal lamellar grafting, compared with scleral grafting. In addition, the corneal grafts maintained their original structure of the graft-recipient complex at 3 weeks postoperatively, and they had a compact stromal layer with a homogenous graft thickness, unlike scleral grafts, suggesting the possible long-term durability of the graft. For these reasons, we suggest that the preserved corneal lamellar tissue may be a favorable alternative as a tectonic graft material in scleral melting diseases, including scleromalacia perforans or necrotizing scleritis. In fact, the lamellar corneal grafting had been attempted in patients with severe scleral melting after pterygium excision in Singapore.[@bib18] It was effective; however, they did not use preserved corneas but rather fresh corneas in unscheduled surgeries. This is because whole-eye donations are not accepted in their country owing to religious reasons, so only in situ corneal removals were performed. Moreover, the study was not a controlled trial, so the effect of tectonic lamellar corneal grafting was not compared with scleral grafting. Afterward, a few studies reported cases with lamellar grafting surgery using preserved corneas in patients with scleral necrosis after radiotherapy or pterygium excision or during limbal dermoid surgery.[@bib19]^--^[@bib22] Moreover, a recent study analyzed the surgical effect of the irradiated corneal versus scleral patch on the rates of erosion in glaucoma drainage device surgery. Although there was no erosion event in either group, the corneal graft was greatly to be preferred.[@bib23] Despite these interests, to the best of our knowledge, there has been no interventional study designed to investigate the comparative effects of tectonic lamellar corneal grafting and scleral grafting on wound healing and inflammation. In general, intact basement membranes are of great importance in enhancing reepithelization by enabling the proliferation of viable epithelial cells at the margins of wounds. Moreover, corneal basement membrane proteins are known to play roles in epithelial differentiation and polarization that could affect antimicrobial barrier function as well as epithelial growth.[@bib24] It is, therefore, possible to speculate that the friable epithelium seen at places in a few cases after scleral grafting could be attributed to the absence of basement membrane at the area of the wound. In addition, the irregular and rough surfaces of scleral grafts without basement membranes might have attracted mucin and detached conjunctival epithelial cells to produce filaments, as shown in this study. In this regard, it would be better to incorporate the additional AM transplantation over scleral grafts to serve as a basement membrane, in so far as preserved scleras are used for tectonic lamellar grafting in scleras.[@bib9]^,^[@bib25]^,^[@bib26] However, this requires additional medical expenses[@bib27] and extra surgical time for surgeons. It is not clear whether the enhanced vascularization over and adjacent to the scleral grafts in our results is the product of healthy wound healing or of an inflammatory reaction. CD34 has been regarded as being ubiquitously related to hematopoietic cells, and accumulating evidence has demonstrated that CD34 expression is present on several other cell types, including vascular endothelial progenitor cells and fibroblastic cells for angiogenesis.[@bib28]^--^[@bib30] The number of CD3**^+^** T cells or CD34**^+^**cells was not significantly different between the two groups, probably owing to the effect of biased sampling, because the focused areas for immunophenotyping were definitely inflamed in both groups. In addition, there is a possibility that a 3-week period may be short to provoke strong activation of adaptive immune system; therefore, the recruited inflammatory cells near grafts might be composed of innate immune cells other than T lymphocytes. However, the colocalization of CD3**^+^** T cells and CD34**^+^**cells in both corneal and scleral grafts suggests that neovascularization may be accompanied by inflammation. Similarly, a previous report supported that inflammation accompanies lymphangiogenesis or vasculogenesis in dry eye.[@bib31] Although more inflammation was observed in the scleral grafting group than in the corneal grafting group, the neovascularization index did not show a statistically significant difference, although a higher tendency toward neovascularization was seen in the scleral grafting group. Given that neovascularization is a phenomenon that may have high intercase variation, it is thought that small-sized analyses have a decreased ability to detect statistical significance. Moreover, neovascularization at the inner layer of opaque scleral grafts might have been omitted from analyses, because photodocumented neovascularization mostly reflects surface area. Scleral melting is a clinically serious problem because it threatens the integrity of the eye. Necrotizing scleritis may occur after infection or the exposure to chemicals, beta irradiation or mitomycin C use during surgery, and may frequently be associated with underling systemic diseases such as rheumatoid arthritis, systemic lupus erythematosus and Wegener\'s granulomatosis.[@bib32] The intrinsic melting tendency in those patients with immune-related vasculitis may continue after grafting surgery[@bib33]; thus, the necessity for more durable and less inflammatogenic graft material can be an important issue. Accordingly, compact collagen fibers in the corneal stromal layer may be another advantage over sclera with regard to the preservation of shapes of the graft structure. In contrast, we sometimes face suture loosening, graft melting, inflammation, and delayed epithelization after lamellar scleral grafting in patients with necrotizing scleritis or scleromalacia. We believe that the irregular and loosened arrangement of scleral stromal collagen fibers and the lack of basement membrane may be linked to the drawbacks of lamellar scleral grafting according to the present study. In our experience, we performed a tectonic lamellar corneal graft using a preserved human cornea successfully in a 73-year-old woman who underwent pterygium excision 15 years ago and recently suffered from scleromalacia with exposed calcium deposits ([Supplementary Fig. S1](#tvst-9-7-38_s001){ref-type="supplementary-material"}). The surface of the graft became rapidly epithelized by more than 80% within 1 week; moreover, there was no observed vascular engorgement in the perigraft area. The preserved cornea is easy to store and is available from several eye banks. Given that the preserved lamellar corneal grafting can be applied when the transparency is required in surgical wounds such as postuveal melanoma excision and when a firm graft is needed, it merits surgical usefulness. However, the cost of the preserved corneal tissue is high (up to \$2,000 USD) and even much higher than preserved sclera and pericardium tissues. A cost-effectiveness analysis would be required in the future study. Our preclinical study is limited by a small number of subjects and lacks a group combining scleral graft with AM transplantation or conjunctival flap. Therefore, the study results might not represent real clinical scenario where scleral grafting is often accompanied by conjunctival or AM grafting. Nevertheless, this is the first comparative study on graft materials between preserved corneas and scleras in scleral defect models. Our results may provide the proof of principle for the future use of preserved corneal lamellar grafts in patients with scleromalacia or necrotizing scleritis. Supplementary Material ====================== ###### Supplement 1 ###### Supplement 2 This study was supported by Seoul National University Hospital Research Fund (grant number: 03-2019-0050). Disclosure: **K.W Kim**, None; **J.S. Ryu**, None; **J.Y. Kim**, None; **M.K. Kim**, None

O God, who adorned Saint Peter Julian Eymard with a wonderful love for the sacred mysteries of the Body and Blood of your Son, graciously grant, that we, too, may be worthy to receive the delights he drew from this divine banquet. Saints are friends of saints, and saints beget saints: Eymard was a friend and contemporary of saints Peter Chanel, Marcellin Champagnat, and Blessed Basil Moreau. He died at the age of fifty-seven in La Mure on 1 August 1868. At his canonization, Saint John XXIII said this of Saint Peter Julian: It is also very fitting that the sacred ceremony occurs during the Second Vatican Ecumenical Council which has as its special purpose to see to it that the pearls of holiness belonging to the crown which encircles the head of the Church should sparkle and shine ever more and more. This extensive gathering of her holy shepherds united with the infallible successor of St Peter not only proposes and reaffirms once again the unchangeable truths left by the divine Master, but also clearly urges that daily, more and more, there be used those holy helps which make us possessors and sharers of divine grace. Furthermore, she enjoins on her children precepts designed to make the Christian way of life better lived. The Council can therefore be said to have no other purpose than to show that here below, the Spouse of Christ possesses every kind of holiness both in deeds, in words, and in spiritual gifts of every kind; that here below she inspires her sons with that holy purpose of the Church expressed so clearly by the Redeemer of the human race: ‘Be perfect as your heavenly Father is perfect’ (Matt., 5: 48). Once these things are understood, it is easy to see that Christians should glory in having such a mother whom everyone ought to admire because of her incredible beauty, divinely infused. Her grandeur does not shine because of gems or pearls that can be seen by human eyes, but rather glows in the splendour and grace which derive from the blood of her Founder and the marvellous virtue of many of her children. As a result, whoever calls himself a Christian ought to observe a way of life which in no way detracts from the supreme honour of their mother and which is not foreign to her precepts and teachings. No one can truly say that he loves his mother who is not afraid of dishonouring her beauty, even a little, by his way of life. The Eucharist, Source of Sanctity Eucharistic life: The Holy Eucharist is the source and the nourishment of all sanctity. Our Predecessor, St Leo the Great, expressed this when he said: ‘The participation in the Body and Blood of Jesus Christ has no other effect than to transform us into Him whom we receive.’ How visible is this progressive transformation into the very life of the divine Saviour, in the admirable development of the virtues of the saints canonised today! And what dealings of particular intimacy with Jesus Eucharistic do we not discover in their ascent to sanctity! The name of Peter Julian suffices to unveil to our eyes the splendid eucharistic triumphs to which, in spite of trials and difficulties of all kinds, he wanted to consecrate his life which prolongs itself in the family founded by him. This little child of five who was found on the altar, his forehead resting on the little door, was the same person who in time would found the Congregation of the Fathers of the Blessed Sacrament and that of the Servants of the Blessed Sacrament, and who would radiate into innumerable armies of priest adorers, his love and tenderness for Christ living in the Eucharist… Marian Piety: At the side of Jesus there stands His Mother, the Queen of all the Saints, the source of sanctity in the Church of God and the first flower of its grace. Intimately associated with the redemption in the eternal plans of the Most High, the Blessed Virgin, as Severiano di Gabala expressed it in song, ‘is the Mother of salvation, the source of light become visible’. Hence filial piety is pleased to consider her at the beginning of all Christian life to ensure its harmonious development and to crown its fullness by her maternal presence. Thus it is not surprising to meet the Blessed Virgin Mary in the life of the three new confessors whom she accompanies step by step. Saint Julian Eymard proposes her as a model to adorers, invoking her as ‘Our Lady of the Blessed Sacrament’……. ….pastoral radiance – the new saints prove it – can be described as the formation of good priests, with fervent souls of adorers, whose ranks have multiplied throughout the world… a Perfect Adorer of the Blessed Sacrament We now desire to add a word for the French pilgrims who have come to assist at the glorification of St Peter Julian Eymard, priest, confessor, founder of two religious families consecrated to the worship of the Blessed Sacrament. He is a saint with whom We have been familiar for many years, as We said above, when as Apostolic Nuncio to France, Providence granted Us the happy opportunity to visit his native land, La Mure d’Isère, near Grenoble. We saw with Our own eyes the poor bed, the humble dwelling where this faithful imitator of Christ gave up his beautiful soul to God. You can surmise, beloved Sons, with what emotion We recall that memory on this day when it is given Us to confer upon him the honours of canonisation. The body of St Peter Julian Eymard is preserved in Paris: but the saint is also somehow present at Rome, in the person of his sons, the Priests of the Blessed Sacrament; it is also a sweet memory for Us to recall visits that We used to make to their Church of St Claude-des-Bourguignons (San Claudio), to unite Ourselves for a few moments to their silent adorations. Besides St Vincent de Paul, St John Eudes, the Curé of Ars, Peter Julian Eymard takes his place in the ranks of the incomparable glory and honour of the country that witnessed their birth, but whose beneficial influence extends far beyond, namely, to the whole Church. His characteristic distinction, the guiding thought of all his priestly activities, one may say, was the Eucharist: eucharistic worship and apostolate. Here, We would like to stress this fact in the presence of the Priests and of the Servants of the Most Blessed Sacrament, in presence also of the members of an Association which is dear to the heart of the Pope, that of the Priest Adorers assembled at this time in Rome, who have come in great numbers to honour this great friend of the Eucharist. Yes, dear Sons, honour and celebrate with Us him who was so perfect an adorer of the Blessed Sacrament; after his example, always place at the centre of your thoughts, of your affections, of the undertakings of your zeal this incomparable source of all grace: the Mystery of Faith, which hides under its veils the Author Himself of grace, Jesus the Incarnate Word. +++ Eymard is a great teacher for those who want to know more about the Eucharist and to have devotion to the Eucharist. It is this Mystery of the Faith which the Church infallibly teaches us is the center, summit and source of All. About the author Paul A. Zalonski is from New Haven, CT. He is a member of the Fraternity of Communion and Liberation, a Catholic ecclesial movement, and an Oblate of Saint Benedict. Contact Paul at paulzalonski[at]yahoo.com.

Foster–Seeley discriminator The Foster–Seeley discriminator is a common type of FM detector circuit, invented in 1936 by Dudley E. Foster and Stuart William Seeley. The circuit was envisioned for automatic frequency control of receivers, but also found application in demodulating an FM signal. It uses a tuned RF transformer to convert frequency changes into amplitude changes. A transformer, tuned to the carrier frequency, is connected to two rectifier diodes. The circuit resembles a full-wave bridge rectifier. If the input equals the carrier frequency, the two halves of the tuned transformer circuit produce the same rectified voltage and the output is zero. As the frequency of the input changes, the balance between the two halves of the transformer secondary changes, and the result is a voltage proportional to the frequency deviation of the carrier. Foster–Seeley discriminators are sensitive to both frequency and amplitude variations, unlike some detectors. Therefore a limiter amplifier stage must be used before the detector, to remove amplitude variations in the signal which would be detected as noise. The limiter acts as a class-A amplifier at lower amplitudes; at higher amplitudes it becomes a saturated amplifier which clips off the peaks and limits the amplitude. Other types of FM detectors are: Slope detector Ratio detector Quadrature detector Phase-locked loop detector Footnotes External links schematic and operation Category:Communication circuits Category:Demodulation de:Diskriminator#Auswertung frequenzmodulierter Signale

Suspicious incident reported in Rhiwbina South Wales Police are investigating a suspicious incident in Rhiwbina which was reported yesterday afternoon. Two men attended the home of a woman aged in her 90s claiming her driveway needed sealing and offering to drive her to a bank to withdraw money to pay for the work. The woman declined their offer and no work was carried out. Enquiries are continuing to identify the men involved and anyone with information is asked to contact South Wales Police on 101 or Crimestoppers on 0800 555 111 quoting occurrence *322252 Please look out for elderly or vulnerable neighbours, friends and relatives, and speak to them about this recent incident. South Wales would remind householders to: • be cautious when approached by door to door traders who call without an appointment. • always get a couple of quotes before agreeing to have work done. • call South Wales Police on 101 or 999 if you feel bullied or harassed into getting work done. • report suspicious behaviour.

using NHibernate; using Recipes.Immutable; using Recipes.NHibernate.Entities; using Recipes.NHibernate.Models; using System; using System.Collections.Generic; using System.Collections.Immutable; using System.Linq; namespace Recipes.NHibernate.Immutable { public class ImmutableScenario : IImmutableScenario<ReadOnlyEmployeeClassification> { readonly ISessionFactory m_SessionFactory; public ImmutableScenario(ISessionFactory sessionFactory) { m_SessionFactory = sessionFactory; } public int Create(ReadOnlyEmployeeClassification classification) { if (classification == null) throw new ArgumentNullException(nameof(classification), $"{nameof(classification)} is null."); using (var session = m_SessionFactory.OpenSession()) { var temp = classification.ToEntity(); session.Save(temp); session.Flush(); return temp.EmployeeClassificationKey; } } public void Delete(ReadOnlyEmployeeClassification classification) { if (classification == null) throw new ArgumentNullException(nameof(classification), $"{nameof(classification)} is null."); using (var session = m_SessionFactory.OpenSession()) { session.Delete(classification.ToEntity()); session.Flush(); } } public ReadOnlyEmployeeClassification? FindByName(string employeeClassificationName) { using (var session = m_SessionFactory.OpenStatelessSession()) { return session.QueryOver<EmployeeClassification>().Where(e => e.EmployeeClassificationName == employeeClassificationName).List() .Select(x => new ReadOnlyEmployeeClassification(x)).SingleOrDefault(); } } public IReadOnlyList<ReadOnlyEmployeeClassification> GetAll() { using (var session = m_SessionFactory.OpenStatelessSession()) { return session.QueryOver<EmployeeClassification>().List() .Select(x => new ReadOnlyEmployeeClassification(x)).ToImmutableArray(); } } public ReadOnlyEmployeeClassification? GetByKey(int employeeClassificationKey) { using (var session = m_SessionFactory.OpenStatelessSession()) { var result = session.Get<EmployeeClassification>(employeeClassificationKey); return new ReadOnlyEmployeeClassification(result); } } public void Update(ReadOnlyEmployeeClassification classification) { if (classification == null) throw new ArgumentNullException(nameof(classification), $"{nameof(classification)} is null."); using (var session = m_SessionFactory.OpenSession()) { session.Update(classification.ToEntity()); session.Flush(); } } } }