Anisotropic Signal Research Articles

Shape dithering for 3D printing

We present an efficient, purely geometric, algorithmic, and parameter free approach to improve surface quality and accuracy in voxel-controlled 3D printing by counteracting quantization artifacts. Such artifacts arise due to the discrete voxel sampling of the continuous shape used to control the 3D printer, and are characterized by low-frequency geometric patterns on surfaces of any orientation. They are visually disturbing, particularly on small prints or smooth surfaces, and adversely affect the fatigue behavior of printed parts. We use implicit shape dithering, displacing the part's signed distance field with a high-frequent signal whose amplitude is adapted to the (anisotropic) print resolution. We expand the reverse generalized Fourier slice theorem by shear transforms, which we leverage to optimize a 3D blue-noise mask to generate the anisotropic dither signal. As a point process it is efficient and does not adversely affect 3D halftoning. We evaluate our approach for efficiency, geometric accuracy and show its advantages over the state of the art.

Testing cosmic anisotropy with the Ep−Eiso (‘Amati’) correlation of GRBs

ABSTRACT We test the possible cosmic anisotropy in 118 long GRBs with the Ep−Eiso (‘Amati’) correlation by employing the dipole fitting (DF) and hemisphere comparison (HC) methods. The distribution of the GRB sample is nearly homogeneous in the sky. The dipole anisotropy is weak in the dipole-modulated $\rm \Lambda$CDM model and the Finslerian cosmological model. The dipole directions from the GRB sample are consistent with ones given by the Pantheon SNe-Ia sample, but with smaller uncertainties. We also investigate whether the GRB sample can reduce the anisotropic signal from inhomogeneous samples like the Pantheon one. The GRB sample is then combined with the Pantheon one, thus providing an SN-G sample. In the dipole-modulated $\rm \Lambda$CDM model, the dipole direction in the SN-G sample shows a considerable change from the one in the Pantheon sample. The angle between the two directions is 26${_{.}^{\circ}}$78. For the HC method, the result of maximum anisotropy level from the G-SN sample is 0.257 ± 0.060 at 68 per cent confidence level (CL) and the corresponding preferred direction is $(l,b)=(82{_{.}^{\circ}}97^{+52{_{.}^{\circ}}73}_{~-61{_{.}^{\circ}}88}, -15{_{.}^{\circ}}09^{+60{_{.}^{\circ}}09}_{~-13{_{.}^{\circ}}54})$. The statistical significance of the $\rm AL_{max}$ is 1.4σ. The angle between the preferred direction and the one from the Pantheon sample is 44${_{.}^{\circ}}$40. Although the amount of data in the GRB sample is about a tenth of that in the Pantheon sample, the GRB sample can considerably impact the results from the Pantheon sample. Our results show that GRBs have the potential to search for a convincing cosmic anisotropy.

Anisotropic Radiation in Heterostructured "Emitter in a Cavity" Nanowire.

Tailorable synthesis of axially heterostructured epitaxial nanowires (NWs) with a proper choice of materials allows for the fabrication of novel photonic devices, such as a nanoemitter in the resonant cavity. An example of the structure is a GaP nanowire with ternary GaPAs insertions in the form of nano-sized discs studied in this work. With the use of the micro-photoluminescence technique and numerical calculations, we experimentally and theoretically study photoluminescence emission in individual heterostructured NWs. Due to the high refractive index and near-zero absorption through the emission band, the photoluminescence signal tends to couple into the nanowire cavity acting as a Fabry–Perot resonator, while weak radiation propagating perpendicular to the nanowire axis is registered in the vicinity of each nano-sized disc. Thus, within the heterostructured nanowire, both amplitude and spectrally anisotropic photoluminescent signals can be achieved. Numerical modeling of the nanowire with insertions emitting in infrared demonstrates a decay in the emission directivity and simultaneous rise of the emitters coupling with an increase in the wavelength. The emergence of modulated and non-modulated radiation is discussed, and possible nanophotonic applications are considered.

Anisotropic Multimessenger Signals from Black Hole Neutrino-dominated Accretion Flows with Outflows in Binary Compact Object Mergers

Abstract A black hole (BH) hyperaccretion system might be born after the merger of a BH and a neutron star (NS) or a binary NS (BNS). In the case of a high mass accretion rate, the hyperaccretion disk is in a state of neutrino-dominated accretion flow (NDAF) and emits numerous anisotropic MeV neutrinos. Only a small fraction of these neutrinos annihilates in the space outside of the disk and then launches ultrarelativistic jets that break away from the merger ejecta to power gamma-ray bursts. Mergers and their remnants are generally considered sources of gravitational waves (GWs), neutrinos, and kilonovae. Anisotropic neutrino emission and anisotropic high-velocity material outflows from central BH–NDAF systems can also trigger strong GWs and luminous disk-outflow-driven (DOD) kilonovae, respectively. In this paper, the anisotropic multimessenger signals from NDAFs with outflows, including DOD kilonovae, MeV neutrinos, and GWs, are presented. According to the results, the typical AB magnitude of the DOD kilonovae is lower than that of astronomical transient AT 2017gfo at the same distance, and it decreases with increasing viewing angles and its anisotropy is not sensitive to the outflow mass distribution but mainly determined by the velocity distribution. Since neutrinos with ≳10 MeV are mainly produced in the inner region of the disk, they will be dramatically deflected to a large viewing angle by relativity effects. Moreover, the strains of GWs induced by anisotropic neutrinos increase with increasing viewing angles. The accumulation of multimessenger detection of the BNS/BH–NS mergers with different viewing angles might further verify the existence of NDAFs with outflows.

Magnetically Induced Anisotropies in the Arrival Directions of Ultra-high-energy Cosmic Rays from Nearby Radio Galaxies

Abstract Detailed simulations of the arrival directions of ultra-high-energy cosmic rays are performed under the assumption of strong and structured extragalactic magnetic field (EGMF) models. Particles leaving Centaurus A, Virgo A, and Fornax A are propagated to Earth, and the simulated anisotropic signal is compared to the dipole and hotspots published by the Pierre Auger and Telescope Array Collaborations. The dominance of the EGMF structure in the arrival directions of events generated in local sources is shown. The absence of events from the Virgo A direction is related to the strong deviation caused by the EGMF. Evidence that these three sources contribute to an excess of events in the direction of the three detected hotspots is presented. Under the EGMF considered here, M82 is shown to have no contribution to the hotspot measured by the Telescope Array Observatory.

Redshift drift cosmography for model-independent cosmological inference

We develop a cosmographic framework for analysing redshift drift signals of nearby sources model-independently, i.e., without making assumptions about the metric description of the Universe. We show that the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) prediction is altered nontrivially by regional anisotropies and inhomogeneities. In particular, we find that the position drift of the sources is nontrivially linked to the redshift drift signal. The redshift drift signal for nearby sources might be formulated in terms of an effective deceleration parameter, which reduces to the FLRW deceleration parameter in the homogeneous and isotropic limit. The presented cosmographic framework can be used for model-independent data analysis, exploiting the fact that the exact anisotropic redshift drift signal at lowest order in redshift is given by a finite set of physically interpretable coefficients. We discuss physical limits of interest as well as challenges related to the framework.

Probing the anisotropic distribution of baryon matter in the Universe using fast radio bursts * *Supported by the National Natural Science Foundation of China (11603005, 11775038, 12005184)

We propose that fast radio bursts (FRBs) can be used as probes to constrain the possible anisotropic distribution of baryon matter in the Universe. Monte Carlo simulations show that 400 (800) FRBs are sufficient to detect the anisotropy at a 95% (99%) confidence level if the dipole amplitude has an order of magnitude of 0.01. However, more FRBs are required to tightly constrain the dipole direction. Even 1000 FRBs are insufficient to constrain the dipole direction within the angular uncertainty at a 95% confidence level. The uncertainty on the dispersion measure of a host galaxy does not significantly affect the results. However, if the dipole amplitude is in the region of 0.001, 1000 FRBs are not enough to correctly detect the anisotropic signal.

Predictions for anisotropic X-ray signatures in the circumgalactic medium: imprints of supermassive black hole driven outflows

ABSTRACT The circumgalactic medium (CGM) encodes signatures of the galaxy-formation process, including the interaction of galactic outflows driven by stellar and supermassive black hole (SMBH) feedback with the gaseous halo. Moving beyond spherically symmetric radial profiles, we study the angular dependence of CGM properties around z = 0 massive galaxies in the IllustrisTNG simulations. We characterize the angular signal of density, temperature, and metallicity of the CGM as a function of galaxy stellar mass, halo mass, distance, and SMBH mass, via stacking. TNG predicts that the CGM is anisotropic in its thermodynamical properties and chemical content over a large mass range, $M_*\sim 10^{10-11.5}\, \mathrm{M}_\odot$. Along the minor axis directions, gas density is diluted, whereas temperature and metallicity are enhanced. These feedback-induced anisotropies in the CGM have a magnitude of 0.1−0.3 dex, extend out to the halo virial radius, and peak at Milky Way-like masses, $M_*\sim 10^{10.8}\, \mathrm{M}_\odot$. In TNG, this mass scale corresponds to the onset of efficient SMBH feedback and the production of strong outflows. By comparing the anisotropic signals predicted by TNG versus other simulations – Illustris and EAGLE – we find that each simulation produces distinct signatures and mass dependencies, implying that this phenomenon is sensitive to the underlying physical models. Finally, we explore X-ray emission as an observable of this CGM anisotropy, finding that future X-ray observations, including the eROSITA all-sky survey, will be able to detect and characterize this signal, particularly in terms of an angular modulation of the X-ray hardness.

Determination of spin chirality using x-ray magnetic circular dichroism

A 3-fold symmetric kagome lattice that has negative spin chirality can give a non-zero x-ray magnetic circular dichroism (XMCD) signal, despite that the total spin moment amounts to zero. This is explained by a hitherto unnoticed rule for the rotational symmetry invariance of the XMCD signal. A necessary condition is the existence of an anisotropic XMCD signal for the local magnetic atom, which can arise from a spin anisotropy either in the ground state or the final state. The angular dependence of the XMCD as a function of the beam direction has an unusual behavior. The maximum dichroism is not aligned along the spin direction, but depends on the relative orientation of the spin with respect to the atomic orientation. Therefore, different geometries can result in the same angular dependence, and the spin direction can only be determined if the atomic orientation is known. The consequences for the x-ray magneto-optical sum rules are given. The integrated XMCD signals are proportional to the anisotropy in the orbital moment and the magnetic dipole term, where the isotropic spin moment drops out.

Development of Highly Sensitive Raman Spectroscopy for Subnano and Single-Atom Detection

Direct detection and characterisation of small materials are fundamental challenges in analytical chemistry. A particle composed of dozens of metallic atoms, a so-called subnano-particle (SNP), and a single-atom catalyst (SAC) are ultimate analysis targets in terms of size, and the topic is now attracting increasing attention as innovative frontier materials in catalysis science. However, characterisation techniques for the SNP and SAC adsorbed on substrates requires sophisticated and large-scale analytical facilities. Here we demonstrate the development of an ultrasensitive, laboratory-scale, vibrational spectroscopic technique to characterise SNPs and SACs. The fine design of nano-spatial local enhancement fields generated by the introduction of anisotropic stellate-shaped signal amplifiers expands the accessibility of small targets on substrates into evanescent electromagnetic fields, achieving not only the detection of isolated small targets but also revealing the effects of intermolecular/interatomic interactions within the subnano configuration under actual experimental conditions. Such a development of “in situ subnano spectroscopy” will facilitate a comprehensive understanding of subnano and SAC science.

Joint analysis of anisotropic power spectrum, bispectrum and trispectrum: application to N-body simulations

We perform for the first time a joint analysis of the monopole and quadrupoles for power spectrum, bispectrum and integrated trispectrum (i-trispectrum) from the redshift space matter field in N-body simulations. With a full Markov Chain Monte Carlo exploration of the posterior distribution, we quantify the constraints on cosmological parameters for an object density of n p = 5 × 10-4 (h Mpc-1)3, redshift z = 0.5, and a covariance corresponding to a survey volume of V survey = 25 (h -1Gpc)3, a set up which is representative of forthcoming galaxy redshift surveys. We demonstrate the complementarity of the bispectrum and i-trispectrum in constraining key cosmological parameters. In particular, compared to the state-of-the-art power spectrum (monopole plus quadrupole) and bispectrum (monopole) analyses, we find 1D 68% credible regions smaller by a factor of (72%,78%,72%,47%,46%) for the parameters (f,σ8,f nl,α∥,α⊥) respectively. This work motivates the additional effort necessary to include the redshift-space anisotropic signal of higher-order statistics in the analysis and interpretation of ongoing and future galaxy surveys.

The competing effects of olivine and orthopyroxene CPO on seismic anisotropy

Orthopyroxene is the second most abundant mineral in the upper mantle and, like olivine, can form crystallographic preferred orientations (CPO) during mantle deformation that may influence mantle seismic anisotropy. Here we use a compilation of 21 peridotites in which both olivine and orthopyroxene CPOs are preserved to document the range of orthopyroxene CPO types and to explore their effects on seismic anisotropy of peridotites. In addition to four previously published orthopyroxene CPO types, this study establishes three new CPO types, two of which result in orthopyroxene fast axes aligned parallel to the lineation and the direction of mantle shear/flow. We calculate bulk elastic tensors for each sample from the measured olivine and orthopyroxene CPOs, and use those in turn to calculate Vp and Vs velocities and anisotropy percentages. In order to explore the relative influence of olivine and orthopyroxene on anisotropy, these properties were calculated for a range of possible modal percentages, from 100% olivine to 100% orthopyroxene. While the effect of orthopyroxene CPO appears to be a negligible influence on the orientation of bulk seismic fast axes, the presence of increasing orthopyroxene decreases the anisotropic signal overall. However, at very low olivine modal percentages, olivine and orthopyroxene can be mutually destructive, where the addition of olivine to a mostly orthopyroxene rock can actually decrease anisotropy.

Localized Anisotropy in the Mantle Transition Zone Due to Flow Through Slab Gaps

AbstractMeasurement of anisotropy advances our understanding of mantle dynamics by linking remote seismic observations to local deformation state through constraints from mineral physics. The Pacific Northwest records the largest depth‐integrated anisotropic signals across the western United States but the depths contributing to the total signal are unclear. We used the amplitudes of orthogonally polarized P‐to‐S converted phases from the mantle transition zone boundaries to identify anisotropy within the ∼400–700 km deep layer. Significant anisotropy is found near slab gaps imaged by prior tomography. Focusing of mantle flow through slab gaps may lead to locally elevated stress that enhances lattice preferred orientation of anisotropic minerals within the transition zone, such as wadsleyite.

Characterization of anisotropic T2W signals from human knee femoral cartilage: The magic angle effect on a spherical surface.

The aim of the current study was to propose a generalized magic angle effect (gMAE) function for characterizing anisotropic T2W signals of human knee femoral cartilage with a spherical surface in clinical studies. A gMAE model function f(α, ε) was formulated for an orientation-dependent (ε) transverse T2 (i.e., 1/R2 ) relaxation in cartilage assuming an axially symmetric distribution (α) of collagen fibers. T2W sagittal images were acquired on an adult volunteer's healthy knee at 3T, and ROI-based average signals S(ε) were extracted from angularly and radially segmented femoral cartilage. Compared with the standard MAE (sMAE) functions in the deep (DZ, α= 0°) and in the superficial (SZ, α= 90°) zones, a general form of R2 orientation-dependent function f(α, ε) was fitted to S(ε), including an isotropic R2 contribution (internal reference [REF]). Goodness of fit was evaluated by root-mean-square deviations (RMSDs). AnF-test and a paired t-test were respectively used to assess significant differences between the observed variances and means, with statistical significance set to pless than.05. As a symmetric orientation-dependence function with a varying dynamic range, the proposed gMAE model outperformed the previous sMAE functions manifested by significantly reduced RMSDs in the DZ (0.239 ± 0.122 vs. 0.267 ± 0.097, p =.014) and in the SZ (0.183 ± 0.081 vs. 0.254 ± 0.085, p <.001). The fitted average angle α (38.5 ± 34.6° vs. 45.1 ± 30.1°, p <.43) and REF (5.092 ± 0.369 vs. 5.305 ± 0.440, p <.001) were smaller in the DZ than those in SZ, in good agreement with the reported collagen fibril microstructural configurations and the nonbound water contribution to R2 in articular cartilage. In conclusion, a general form of the magic angle effect function was proposed and demonstrated for better characterizing anisotropic T2W signals from human knee femoral cartilage at 3T in clinical studies.

Anisotropic conductive reduced graphene oxide/silk matrices promote post-infarction myocardial function by restoring electrical integrity

Myocardial infarction (MI) remains the leading cause of death globally, often leading to impaired cardiac function and pathological myocardial microenvironment. Electrical conduction abnormalities of the infarcted myocardium not only induce adverse myocardial remodeling but also prevent tissue repair. Restoring the myocardial electrical integrity, particularly the anisotropic electrical signal propagation within the injured area after infarction is crucial for an effective function recovery. Herein, optimized reduced graphene oxide (rGO) functionalized electrospun silk fibroin (rGO/silk) biomaterials presenting anisotropic conductivity and enhanced suturablity were developed and investigated as cardiac patches for their potential in improving the post-MI myocardial function of rat models. The results show that the anisotropic conductive rGO/silk patches exhibit remarkable therapeutic effect on repairing the infarcted myocardium compared to the nonconductive silk and isotropic conductive rGO/silk patches as determined by the enhanced pumping function, reduced susceptibility to arrhythmias, thickened left ventricular walls and improved survival of functional cardiomyocytes. Their notable effect on promoting the angiogenesis of capillaries in the infarcted myocardium has also been demonstrated. This study highlights an effective and biomimetic reconstruction of the electrical myocardial microenvironment based on the anisotropic conductive rGO/silk biomaterials as a promising option for promoting the repair of infarcted myocardium. Statement of significanceThe dysfunctional electrical microenvironment in the infarcted myocardium not only aggravates the adverse myocardial remodeling but also limits the effect of cardiac regenerative medicine. Although various conductive biomaterials have been employed to restore the electrical network in the infarcted myocardium in vivo, the anisotropic nature of the myocardial electrical microenvironment which enables directional electrical signal propagation were neglected. In this study, an anisotropic conductive rGO/silk biomaterial system is developed to improve the myocardial function post infarction by restoring the anisotropic electrical microenvironment in the infarcted myocardium. The promoted effects of anisotropic conductive grafts on repairing infarcted hearts are demonstrated with improved pumping function, cardiomyocyte survival, resistance to ventricular fibrillation, and angiogenesis of capillary network.

ET sensitivity to the anisotropic Stochastic Gravitational Wave Background

We study the sensitivity of a pair of Einstein Telescopes (ET) (hypothetically located at the two sites currently under consideration for ET) to the anisotropies of the Stochastic Gravitational Wave Background (SGWB). We focus on the ℓ =0,2,4 multipoles of an expansion of the SGWB in spherical harmonics, since the sensitivity to other multipoles is suppressed due to the fact that this pair of detector operates in a regime for which the product between the observed frequency and the distance between the two sites is much smaller than one. In this regime, the interferometer overlap functions for the anisotropic signal acquire very simple analytic expressions. These expressions can also be applied to any other pairs of interferometers (each one of arbitrary opening angle between its two arms) operating in this regime. Once the measurements at the vertices of the two sites are optimally combined, the sensitivity to the multipoles of the SGWB depends only on the latitude of the two sites, on the difference of their longitude, but not on the orientation of their arms.

Quantifying the hydroxyapatite orientation near the ossification front in a piglet femoral condyle using X-ray diffraction tensor tomography

While a detailed knowledge of the hierarchical structure and morphology of the extracellular matrix is considered crucial for understanding the physiological and mechanical properties of bone and cartilage, the orientation of collagen fibres and carbonated hydroxyapatite (HA) crystallites remains a debated topic. Conventional microscopy techniques for orientational imaging require destructive sample sectioning, which both precludes further studies of the intact sample and potentially changes the microstructure. In this work, we use X-ray diffraction tensor tomography to image non-destructively in 3D the HA orientation in a medial femoral condyle of a piglet. By exploiting the anisotropic HA diffraction signal, 3D maps showing systematic local variations of the HA crystallite orientation in the growing subchondral bone and in the adjacent mineralized growth cartilage are obtained. Orientation maps of HA crystallites over a large field of view (~ 3 × 3 × 3 mm3) close to the ossification (bone-growth) front are compared with high-resolution X-ray propagation phase-contrast computed tomography images. The HA crystallites are found to predominantly orient with their crystallite c-axis directed towards the ossification front. Distinct patterns of HA preferred orientation are found in the vicinity of cartilage canals protruding from the subchondral bone. The demonstrated ability of retrieving 3D orientation maps of bone-cartilage structures is expected to give a better understanding of the physiological properties of bones, including their propensity for bone-cartilage diseases.

Constraining primordial non-Gaussianity with postreconstructed galaxy bispectrum in redshift space

Galaxy bispectrum is a promising probe of inflationary physics in the early Universe as a measure of primordial non-Gaussianity (PNG), whereas its signal-to-noise ratio is significantly affected by the mode coupling due to nonlinear gravitational growth. In this paper, we examine the standard reconstruction method of linear cosmic mass density fields from nonlinear galaxy density fields to decorrelate the covariance in redshift-space galaxy bispectra. In particular, we evaluate the covariance of the bispectrum for massive-galaxy-sized dark matter halos with reconstruction by using 4000 independent $N$-body simulations. Our results show that the bispectrum covariance for the postreconstructed field approaches the Gaussian prediction at scale of $k&lt;0.2\text{ }\text{ }h\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. We also verify the leading-order PNG-induced bispectrum is not affected by details of the reconstruction with perturbative theory. We then demonstrate the constraining power of the postreconstructed bispectrum for PNG at redshift of approximately 0.5. Further, we perform a Fisher analysis to make a forecast of PNG constraints by galaxy bispectra including anisotropic signals. Assuming a massive galaxy sample in the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey, we find that the postreconstructed bispectrum can constrain the local, equilateral, and orthogonal types of PNG with $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\sim}13$, 90, and 42, respectively, improving the constraints with the prereconstructed bispectrum by a factor of 1.3--3.2. In conclusion, the reconstruction plays an essential role in constraining various types of PNG signatures with a level of $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\lesssim}1$ from the galaxy bispectrum based on upcoming galaxy surveys.

Split Ga vacancies and the unusually strong anisotropy of positron annihilation spectra in β−Ga2O3

We report a systematic first principles study on positron annihilation parameters in the $\beta$-Ga$_2$O$_3$ lattice and Ga mono-vacancy defects complemented with orientation-dependent experiments of the Doppler broadening of the positron-electron annihilation. We find that both the $\beta$-Ga$_2$O$_3$ lattice and the considered defects exhibit unusually strong anisotropy in their Doppler broadening signals. This anisotropy is associated with low symmetry of the $\beta$-Ga$_2$O$_3$ crystal structure that leads to unusual kind of one-dimensional confinement of positrons even in the delocalized state in the lattice. In particular, the split Ga vacancies recently observed by scanning transmission electron microscopy produce unusually anisotropic positron annihilation signals. We show that in experiments, the positron annihilation signals in $\beta$-Ga$_2$O$_3$ samples seem to be often dominated by split Ga vacancies.

Review and experimental verification of x-ray dark-field signal interpretations with respect to quantitative isotropic and anisotropic dark-field computed tomography

Talbot(-Lau) interferometric x-ray and neutron dark-field imaging has, over the past decade, gained substantial interest for its ability to provide insights into a sample’s microstructure below the imaging resolution by means of ultra small angle scattering effects. Quantitative interpretations of such images depend on models of the signal origination process that relate the observable image contrast to underlying physical processes. A review of such models is given here and their relation to the wave optical derivations by Yashiro et al and Lynch et al as well as to small angle scattering is discussed. Fresnel scaling is introduced to explain the characteristic distance dependence observed in cone beam geometries. Moreover, a model describing the anisotropic signals of fibrous objects is derived. The Yashiro-Lynch model is experimentally verified both in radiographic and tomographic imaging in a monochromatic synchrotron setting, considering both the effects of material and positional dependence of the resulting dark-field contrast. The effect of varying sample–detector distance on the dark-field signal is shown to be non-negligible for tomographic imaging, yet can be largely compensated for by symmetric acquisition trajectories. The derived orientation dependence of the dark-field contrast of fibrous materials both with respect to variations in autocorrelation width and scattering cross section is experimentally validated using carbon fiber reinforced rods.