Gaussian Ring Research Articles

Plenoptic particle image velocimetry with multiple plenoptic cameras

Plenoptic particle image velocimetry was recently introduced as a viable three-dimensional, three-component velocimetry technique based on light field cameras. One of the main benefits of this technique is its single camera configuration allowing the technique to be applied in facilities with limited optical access. The main drawback of this configuration is decreased accuracy in the out-of-plane dimension. This work presents a solution with the addition of a second plenoptic camera in a stereo-like configuration. A framework for reconstructing volumes with multiple plenoptic cameras including the volumetric calibration and reconstruction algorithms, including: integral refocusing, filtered refocusing, multiplicative refocusing, and MART are presented. It is shown that the addition of a second camera improves the reconstruction quality and removes the ‘cigar’-like elongation associated with the single camera system. In addition, it is found that adding a third camera provides minimal improvement. Further metrics of the reconstruction quality are quantified in terms of a reconstruction algorithm, particle density, number of cameras, camera separation angle, voxel size, and the effect of common image noise sources. In addition, a synthetic Gaussian ring vortex is used to compare the accuracy of the single and two camera configurations. It was determined that the addition of a second camera reduces the RMSE velocity error from 1.0 to 0.1 voxels in depth and 0.2 to 0.1 voxels in the lateral spatial directions. Finally, the technique is applied experimentally on a ring vortex and comparisons are drawn from the four presented reconstruction algorithms, where it was found that MART and multiplicative refocusing produced the cleanest vortex structure and had the least shot-to-shot variability. Filtered refocusing is able to produce the desired structure, albeit with more noise and variability, while integral refocusing struggled to produce a coherent vortex ring.

Producing an efficient, collimated, and thin annular beam with a binary axicon.

We propose and demonstrate a method to produce a thin and highly collimated annular beam that propagates similarly to an ideal thin Gaussian ring beam, maintaining its excellent propagation properties. Our optical configuration is composed of a binary axicon, a circular binary phase grating, and a lens, making it robust and well suited for high-power lasers. It has a near-perfect circular profile with a dark center, and its large radius to waist ratio is achieved with high conversion efficiency. The measured profile and propagation are in excellent agreement with a numerical Fourier simulation we perform.

A 1/t algorithm with the density of two states for estimating multidimensional integrals

In this work we developed a 1∕t algorithm for numerical integration in high dimensions. A large amount of computation time is wasted when a random walker is unable to reach a rare state at the sharp peak of an integrand, and becomes trapped after falling through that state. In this study, the density of states was divided into only two levels by sampling an arbitrary point in the range of the integrand, rather than into many levels using a fixed bin width (grid discretization on continuous space). The technique is quite straightforward and easy to implement. It avoids the need to determine the exact boundaries of the integrand, which is often a non-trivial task. Simulations show that our method is able to significantly reduce the number of Monte Carlo trials required, and therefore the simulation time. The potential of the proposed method was demonstrated by application to two multidimensional integrals: the Gaussian ring and the setting sun Feynman diagram. The results confirm that the proposed method can be applied to the calculation of multidimensional integrals without error saturation, yielding accurate values in applications where other numerical methods fail.

Laser beam shaping for enhanced Zero-Group Velocity Lamb modes generation.

Optimization of Lamb modes induced by laser can be achieved by adjusting the spatial source distribution to the mode wavelength (λ). The excitability of Zero-Group Velocity (ZGV) resonances in isotropic plates is investigated both theoretically and experimentally for axially symmetric sources. Optimal parameters and amplitude gains are derived analytically for spot and annular sources of either Gaussian or rectangular energy profiles. For a Gaussian spot source, the optimal radius is found to be λZGV/π. Annular sources increase the amplitude by at least a factor of 3 compared to the optimal Gaussian source. Rectangular energy profiles provide higher gain than Gaussian ones. These predictions are confirmed by semi-analytical simulation of the thermoelastic generation of Lamb waves, including the effect of material attenuation. Experimentally, Gaussian ring sources of controlled width and radius are produced with an axicon-lens system. Measured optimal geometric parameters obtained for Gaussian and annular beams are in good agreement with theoretical predictions. A ZGV resonance amplification factor of 2.1 is obtained with the Gaussian ring. Such source should facilitate the inspection of highly attenuating plates made of low ablation threshold materials like composites.

Filtered refocusing: a volumetric reconstruction algorithm for plenoptic-PIV

A new algorithm for reconstruction of 3D particle fields from plenoptic image data is presented. The algorithm is based on the technique of computational refocusing with the addition of a post reconstruction filter to remove the out of focus particles. This new algorithm is tested in terms of reconstruction quality on synthetic particle fields as well as a synthetically generated 3D Gaussian ring vortex. Preliminary results indicate that the new algorithm performs as well as the MART algorithm (used in previous work) in terms of the reconstructed particle position accuracy, but produces more elongated particles. The major advantage to the new algorithm is the dramatic reduction in the computational cost required to reconstruct a volume. It is shown that the new algorithm takes 1/9th the time to reconstruct the same volume as MART while using minimal resources. Experimental results are presented in the form of the wake behind a cylinder at a Reynolds number of 185.

On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \cdots x_{n+1} \in I$ for $x_1, \ldots, x_{n+1} \in R$, then there are $n$ of the $x_i$'s whose product is in $I$ and conjecture that $\omega_{R[X]}(I[X])=\omega_R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega_R(I)= \min \{n\colon\text{$I$ is an $n$-absorbing ideal of $R$}\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Pr\"ufer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.

Gaussian Ring Basis Functions for the Analysis of Modulated Metasurface Antennas

This paper presents a novel type of basis functions, whose spectral- and space-domain properties can be exploited for the efficient method of moments (MoM) analysis of planar metasurface (MTS) antennas. The effect of the homogenized MTS is introduced in the integral equation as an impedance boundary condition (IBC). The proposed basis functions are shaped as Gaussian-type rings with small width and linear azimuthal phase. The analytical form of the spectrum of the Gaussian ring basis allows for a closed-form evaluation of the MoM impedance matrix’s entries. Moreover, these basis functions account for the global evolution of the surface current density in an effective manner, reducing the size of the MoM system of equations with respect to the case of subdomain basis functions. These features allow one to carry out a direct solution for problems with a diameter of up to 15 wavelengths in less than 1 min using a conventional laptop. The applicability on practical antennas has been tested through the full-wave analysis of MTS antennas implemented with small printed elements.

Reanalysis on coordinate system for Gaussian beam reflection

Some problems and confusions related to matrix analysis for Gaussian beam reflection are clarified. We can choose right-handed coordinates before and after mirror reflection which is called the traditional coordinate system in this Letter. We can also choose a coordinate system with chirality inversion which is called a novel coordinate system in this Letter. The matrix describing optical components should coincide with the selected coordinate system; errors or confusions will appear otherwise. Spherical mirror reflection and coordinate rotation in nonplanar ring resonators are used to clarify some confusion in previous publications due to disorder of the coordinate system. This work is significant to both Gaussian beam propagation analysis and ring laser resonator design.

On the mechanism of the formation and mass growth of solar system bodies at the early stages of evolution

In the present paper, the early evolution of the Solar System with regard to the process of the formation and mass growth of small bodies in protoplanetary rings is considered. This process is modeled with the use of the results that were obtained in the analysis of the libration motions of asteroids. It has been shown that libration centers could occur prior to the formation of the bodies, when the redistribution of gaseous-dust grains occurred in protoplanetary rings. The distribution density of gaseous-dust grains could be similar or equal to that in the Gaussian ring or its analogs for long-period librations. In this case, the bodies can be initially formed and grow in stable libration centers.

Bazzoni–Glaz conjecture

In [2], Bazzoni and Glaz (2007) conjecture that the weak global dimension of a Gaussian ring is 0,1 or ∞. In this paper, we prove their conjecture.

Gaussian Integers

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Potential of a gaussian ring. A new approach

A fundamentally new approach to an elliptic Gaussian ring has been developed. It has been ascertained that it can be produced from a uniform plane elliptic disk by mass balayage into an elementary homothetic layer with the center of homothety at an ellipse focus. An advantage of new interpretation is in the fact that the spatial potential of a Gaussian ring is expressed in terms of the potential of a uniform elliptic disk, well-known in the finite form, and its derivatives. A general formula for the potential of a two-dimensional homothetic layer has been derived with this purpose. As a result, the potential of a Gaussian ring is represent-able in a simple analytical form in terms of standard complete elliptic integrals in both elliptic and Cartesian coordinates. The mass asymmetry along the ring is considered explicitly. The potential formulas are verified numerically and have no singular points at ellipse foci. Particular cases are considered; the 3D potential surface and system of equipotentials are constructed. Knowledge of the potential extends the range of application of a Gaussian ring in the problem of calculation of secular perturbations in celestial mechanics.

Numerical validation of the volume penalization method in three-dimensional pseudo-spectral simulations

The penalization method is used to simulate the effect of boundaries on fluid flows without body-fitted mesh. It is introduced here in a three-dimensional Fourier pseudo-spectral code with periodic boundary conditions. We propose an implicit implementation for the penalization term, thus getting rid of the strong stability constraint stemming from the stiffness of the penalized equations. The convergence of this implicit penalized method is shown by comparison with solutions of Navier–Stokes equations with explicit boundary conditions. Different test-cases are used, among which a comparison with channel flow simulations with a Chebyshev–Fourier pseudo-spectral code in which the projection base automatically ensures the no-slip boundary condition and the divergence-free condition. We consider inertial or internal gravity waves reflection on solid walls, hence assessing the correct account of pressure coupling for wave-propagation and reflection by the implicit penalized algorithm, and the common benchmark of a Gaussian vortex ring impacting on a normal flat plane solid boundary. Numerical convergence is characterized by increasing the grid resolution and by reducing the penalization parameter η.

Self-Consistent Field Theory of Gaussian Ring Polymers

We present a self-consistent field theory (SCFT) formalism of topologically unconstrained ring polymers for the first time. The resulting static properties of homogeneous and inhomogeneous ring polymers are compared with the random phase approximation (RPA) results. For ring homopolymer mixture, as χN increases, the interfacial width and segment profile converge to those of the linear polymer mixture. The critical point for the ring homopolymer system is exactly the same as the linear polymer case, (χN)c = 2, since the critical point does not depend on the local structure of polymers. The critical point for ring diblock copolymer melts is (χN)c = 17.795, which is ∼1.7 times that of linear diblock copolymer melts, (χN)c = 10.495. The difference results from the ring structure constraint.

Force function of a slightly elliptical Gaussian ring and its generalization to a nearly coplanar system of rings

An analytical expression of the force function of a slightly elliptical Gaussian ring is proposed. The obtained power expansion of ellipse eccentricity contains the terms up to the third power inclusive. The coefficients of this expansion depend on the coordinates of the sampling point and semi-major axis of the ellipse. In the advanced application to the system of the major moons of Uranus for the investigation into secular mutual perturbations, the generalization of the force function is performed for the case of a nearly coplanar system of rings. The force function obtained in this work is more accurate than the earlier known ones and includes the terms up to the third power inclusive with respect to the eccentricities and equatorial inclinations of the moons’ orbits.

The JCMT Nearby Galaxies Legacy Survey - V. The CO(J= 3-2) distribution and molecular outflow in NGC 4631

We have made the first map of CO(J= 3–2) emission covering the disc of the edge-on galaxy, NGC 4631, which is known for its spectacular gaseous halo. The strongest emission, which we model with a Gaussian ring, occurs within a radius of 5 kpc. Weaker disc emission is detected out to radii of 12 kpc, the most extensive molecular component yet seen in this galaxy. From comparisons with infrared data, we find that CO(J= 3–2) emission more closely follows the hot dust component, rather than the cold dust, consistent with it being a good tracer of star formation. The first maps of R3−2/1−0, H2 mass surface density and star formation efficiency (SFE) have been made for the inner 2.4 kpc radius region. Only 20 per cent of the star formation occurs in this region and excitation conditions are typical of galaxy discs, rather than of central starbursts. The SFE suggests long gas consumption time-scales (>109 yr). The velocity field is dominated by a steeply rising rotation curve in the region of the central molecular ring followed by a flatter curve in the disc. A very steep gradient in the rotation curve is observed at the nucleus, providing the first evidence for a central concentration of mass: Mdyn= 5 × 107 M⊙ within a radius of 282 pc. The velocity field shows anomalous features indicating the presence of molecular outflows; one of them is associated with a previously observed CO(J= 1–0) expanding shell. Consistent with these outflows is the presence of a thick (z up to 1.4 kpc) CO(J= 3–2) disc. We suggest that the interaction between NGC 4631 and its companion(s) has agitated the disc and also initiated star formation which was likely higher in the past than it is now. These may be necessary conditions for seeing prominent haloes.

The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for $${{\rm PU}(2,1,\mathbb{Z} [i])}$$ , that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers $${\mathbb{Z} [i]}$$ . We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

Imaging the asymmetric dust shell around CI Cam with long baseline optical interferometry

We present the first high angular resolution observation of the B[e] star/X-ray transient object CI Cam, performed with the two-telescope Infrared Optical Telescope Array (IOTA), its upgraded three-telescope version (IOTA3T) and the Palomar Testbed Interferometer (PTI). Visibilities and closure phases were obtained using the IONIC-3 integrated optics beam combiner. CI Cam was observed in the near-infrared H and K spectral bands, wavelengths well suited to measure the size and study the geometry of the hot dust surrounding CI Cam. The analysis of the visibility data over an 8 year period from soon after the 1998 outburst to 2006 shows that the dust visibility has not changed over the years. The visibility data shows that CI Cam is elongated which confirms the disc-shape of the circumstellar environment and totally rules out the hypothesis of a spherical dust shell. Closure phase measurements show direct evidence of asymmetries in the circumstellar environment of CI Cam and we conclude that the dust surrounding CI Cam lies in an inhomogeneous disc seen at an angle. The near-infrared dust emission appears as an elliptical skewed Gaussian ring with a major axis a = 7.58 +/- 0.24 mas, an axis ratio r = 0.39 +/- 0.03 and a position angle theta = 35 +/- 2 deg.

Security of a Class of Knapsack Public-Key Cryptosystems against Low-Density Attack

In 2003, Kobayashi et al. proposed a new class of knapsack public-key cryptosystems over Gaussian integer ring. This scheme using two-sequences as the public key. In 2005, Sakamoto and Hayashi proposed an improved version of Kobayashi's scheme. In this paper, we propose the knapsack PKC using l-sequences as the public key and present the low-density attack on it. We have described Schemes R and G for l = 2, in which the public keys are constructed over rational integer ring and over Gaussian integer ring, respectively. We discusses on the difference of the security against the low-density attack. We show that the security levels of Schemes R and G differ only slightly.

AMBER/VLTI interferometric observations of the recurrent Nova RS Ophiuchii 5.5 days after outburst

Aims.We report on spectrally dispersed interferometric AMBER/VLTI observations of the recurrent nova RS Oph five days after the discovery of its outburst on 2006 Feb. 12. Methods: Using three baselines ranging from 44 to 86 m, and a spectral resolution of λ/δλ=1500, we measured the extension of the milliarcsecond-scale emission in the K band continuum and in the Brγ and He I 2.06 μm lines, allowing us to get an insight into the kinematics of the line forming regions. The continuum visibilities were interpreted by fitting simple geometric models consisting of uniform and Gaussian ellipses, ring and binary models. The visibilities and differential phases in the Brγ line were interpreted using skewed ring models aiming to perform a limited parametric reconstruction of the extension and kinematics of the line forming region. Results: The limited uv coverage does not allow discrimination between filled models (uniform or Gaussian ellipses) and rings. Binary models are discarded because the measured closure phase in the continuum is close to zero. The visibilities in the lines are at a low level compared to their nearby continuum, consistent with a more extended line forming region for He I 2.06 μm than Brγ. The ellipse models for the continuum and for the lines are highly flattened (b/a˜0.6) and share the same position angle (PA˜140°). Their typical Gaussian extensions are 3.1×1.9 mas, 4.9×2.9 mas and 6.3×3.6 mas for the continuum, Brγ and He I 2.06 μm lines, respectively. Two radial velocity fields are apparent in the Brγ line: a slow expanding ring-like structure (v_rad≤1800 km s-1), and a fast structure extended in the E-W direction (v_rad˜ 2500{-}3000 km s-1), a direction that coincides with the jet-like structure seen in the radio. These results confirm the basic fireball model, contrary to the conclusions of other interferometric observations conducted by Monnier et al. (2006a, ApJ, 647, L127). Based on observations collected at the European Southern Observatory, Paranal, Chile, within the programme 276.D-5049.