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Stability is a basic requirement when studying the behavior of dynamical systems. However, stabilizing dynamical systems via reinforcement learning is challenging because only little data can be collected over short time horizons before instabilities are triggered and data become meaningless. This work introduces a reinforcement learning approach that is formulated over latent manifolds of unstable dynamics so that stabilizing policies can be trained from few data samples. The unstable manifolds are minimal in the sense that they contain the lowest dimensional dynamics that are necessary for learning policies that guarantee stabilization. This is in stark contrast to generic latent manifolds that aim to approximate all -- stable and unstable -- system dynamics and thus are higher dimensional and often require higher amounts of data. Experiments demonstrate that the proposed approach stabilizes even complex physical systems from few data samples for which other methods that operate either directly in the system state space or on generic latent manifolds fail., Comment: 29 pages, 10 figures, 1 table

Polytopic autoencoders provide low-dimensional parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it outperforms standard linear approaches in view of LPV approximations of nonlinear systems and how the particular architecture enables higher order series expansions at little extra computational effort. We illustrate the properties and potentials of this approach to computational nonlinear controller design for large-scale systems with a thorough numerical study., Comment: 9 pages, 6 figures, 2 tables

The root mean squared error is an important measure used in a variety of applications such as structural dynamics and acoustics to model averaged deviations from standard behavior. For large-scale systems, simulations of this quantity quickly become computationally prohibitive. Classical model order reduction techniques attempt to resolve this issue via the construction of surrogate models that emulate the root mean squared error measure using an intermediate linear system. However, this approach requires a potentially large number of linear outputs, which can be disadvantageous in the design of reduced-order models. In this work, we consider directly the root mean squared error as the quantity of interest using the concept of quadratic-output models and propose several new model reduction techniques for the construction of appropriate surrogates. We test the proposed methods on a model for the vibrational response of a plate with tuned vibration absorbers., Comment: 8 pages, 2 figures

Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati equation. This is not the case when it comes to large-scale sparse coefficient matrices. In this paper, we provide a reformulation of the Newton-Kleinman iteration scheme for continuous-time algebraic Riccati equations using indefinite symmetric low-rank factorizations. This allows the application of the method to the case of general large-scale sparse coefficient matrices. We provide convergence results for several prominent realizations of the equation and show in numerical examples the effectiveness of the approach., Comment: 29 pages, 1 figure, 10 tables

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the Iterative Rational Krylov Algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures., Comment: 24 pages, 5 figures, 4 tables

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high-dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures., Comment: 29 pages, 7 figures

An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems' transfer functions are rewritten in different barycentric forms using different parametrizations. Loewner-like algorithms are developed for the explicit computation of second-order systems from data based on the developed barycentric forms. Numerical experiments show the performance of these new structured data driven modeling methods compared to other interpolation-based data-driven modeling techniques from the literature., Comment: 27 pages, 5 figures

Nonlinear feedback design via state-dependent Riccati equations is well established but unfeasible for large-scale systems because of computational costs. If the system can be embedded in the class of linear parameter-varying (LPV) systems with the parameter dependency being affine-linear, then the nonlinear feedback law has a series expansion with constant and precomputable coefficients. In this work, we propose a general method to approximating nonlinear systems such that the series expansion is possible and efficient even for high-dimensional systems. We lay out the stabilization of incompressible Navier-Stokes equations as application, discuss the numerical solution of the involved matrix-valued equations, and confirm the performance of the approach in a numerical example., Comment: 8 pages, 3 figures

This work introduces a data-driven control approach for stabilizing high-dimensional dynamical systems from scarce data. The proposed context-aware controller inference approach is based on the observation that controllers need to act locally only on the unstable dynamics to stabilize systems. This means it is sufficient to learn the unstable dynamics alone, which are typically confined to much lower dimensional spaces than the high-dimensional state spaces of all system dynamics and thus few data samples are sufficient to identify them. Numerical experiments demonstrate that context-aware controller inference learns stabilizing controllers from orders of magnitude fewer data samples than traditional data-driven control techniques and variants of reinforcement learning. The experiments further show that the low data requirements of context-aware controller inference are especially beneficial in data-scarce engineering problems with complex physics, for which learning complete system dynamics is often intractable in terms of data and training costs., Comment: 27 pages, 10 figures

In this paper, we consider the structure-preserving model order reduction problem for multi-input/multi-output bilinear control systems by tangential interpolation. We propose a new type of tangential interpolation problem for structured bilinear systems, for which we develop a new structure-preserving interpolation framework. This new framework extends and generalizes different formulations of tangential interpolation for bilinear systems from the literature and also provides a unifying framework. We then derive explicit conditions on the projection spaces to enforce tangential interpolation in different settings, including conditions for tangential Hermite interpolation. The analysis is illustrated by means of three numerical examples., Comment: 33 pages, 9 figures

Robust controllers that stabilize dynamical systems even under disturbances and noise are often formulated as solutions of nonsmooth, nonconvex optimization problems. While methods such as gradient sampling can handle the nonconvexity and nonsmoothness, the costs of evaluating the objective function may be substantial, making robust control challenging for dynamical systems with high-dimensional state spaces. In this work, we introduce multi-fidelity variants of gradient sampling that leverage low-cost, low-fidelity models with low-dimensional state spaces for speeding up the optimization process while nonetheless providing convergence guarantees for a high-fidelity model of the system of interest, which is primarily accessed in the last phase of the optimization process. Our first multi-fidelity method initiates gradient sampling on higher fidelity models with starting points obtained from cheaper, lower fidelity models. Our second multi-fidelity method relies on ensembles of gradients that are computed from low- and high-fidelity models. Numerical experiments with controlling the cooling of a steel rail profile and laminar flow in a cylinder wake demonstrate that our new multi-fidelity gradient sampling methods achieve up to two orders of magnitude speedup compared to the single-fidelity gradient sampling method that relies on the high-fidelity model alone., Comment: 28 pages, 4 figures

Learning controllers from data for stabilizing dynamical systems typically follows a two step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) $n$, then there always exist $n$ states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model., Comment: 29 pages, 4 figures

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We show and compare the effectiveness of all considered methods by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems., Comment: 31 pages, 18 figures

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples., Comment: 19 pages, 2 figures, 5 tables

In this paper, we develop a structure-preserving formulation of the data-driven vector fitting algorithm for the case of modally damped mechanical systems. Using the structured pole-residue form of the transfer function of modally damped second-order systems, we propose two possible structured extensions of the barycentric formula of system transfer functions. Integrating these new forms within the classical vector fitting algorithm leads to the formulation of two new algorithms that allow the computation of modally damped mechanical systems from data in a least squares fashion. Thus, the learned model is guaranteed to have the desired structure. We test the proposed algorithms on two benchmark models., Comment: 8 pages, 2 figures

Output-based controllers are known to be fragile with respect to model uncertainties. The standard $\mathcal{H}_{\infty}$-control theory provides a general approach to robust controller design based on the solution of the $\mathcal{H}_{\infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples., Comment: 20 pages, 4 figures, 3 tables

Suppressing vibrations in mechanical systems, usually described by second-order dynamical models, is a challenging task in mechanical engineering in terms of computational resources even nowadays. One remedy is structure-preserving model order reduction to construct easy-to-evaluate surrogates for the original dynamical system having the same structure. In our work, we present an overview of recently developed structure-preserving model reduction methods for second-order systems. These methods are based on modal and balanced truncation in different variants, as well as on rational interpolation. Numerical examples are used to illustrate the effectiveness of all described methods., Comment: 21 pages, 8 figures

CK Vul is the remnant of an energetic eruption known as Nova 1670 that is thought to be caused by a stellar merger. The remnant is composed of (1) a large hourglass nebula of recombining gas (of 71\arcsec size), very similar to some classical planetary and pre-planetary nebulae (PPNe), and (2) of a much smaller and cooler inner remnant prominent in millimeter-wave emission from molecules. We investigate the three-dimensional spatio-kinematic structure of both components. The analysis of the hourglass structure yields a revised distance to the object of >2.6 kpc, at least 3.7 times greater than so far assumed. At this distance, the stellar remnant has a bolometric luminosity >12 L$_{\odot}$ and is surrounded by molecular material of total mass >0.8 M$_{\odot}$ (the latter value has a large systematic uncertainty). We also analyzed the architecture of the inner molecular nebula using ALMA observations of rotational emission lines obtained at subarcsecond resolution. We find that the distribution of neutral and ionized gas in the lobes can be reproduced by several nested and incomplete shells or jets with different velocity fields and varying orientations. The analysis indicates that the molecular remnant was created in several ejection episodes, possibly involving an interacting binary system. We calculated the linear momentum ($\approx$10$^{40}$ g cm s$^{-1}$) and kinetic energy ($\approx$10$^{47}$ erg) of the CK Vul outflows and find them within the limits typical for classical PPNe. Given the similarities of the CK Vul outflows to PPNe, we suggest there may CK Vul analogs among wrongly classified PPNe with low intrinsic luminosities, especially among PPNe with post-red-giant-branch central stars., Comment: 12 pages, animations at: https://bit.ly/3hR6jSF and https://shorturl.at/gwzOT . Comments welcomed

In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis., Comment: 10 pages, 2 figures

In this paper, we extend the structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples., Comment: 26 pages, 9 figures

For an easy use of model order reduction techniques in applications, software solutions are needed. In this paper, we describe the MORLAB, Model Order Reduction LABoratory, toolbox as an efficient implementation of model reduction techniques for dense, medium-scale linear time-invariant systems. Giving an introduction to the underlying programming principles of the toolbox, we show the basic idea of spectral splitting and present an overview about implemented model reduction techniques. Two numerical examples are used to illustrate different use cases of the MORLAB toolbox., Comment: 17 pages, 6 figures, 5 tables

We perform numerical simulations to investigate the stellar wind from interacting binary stars. Our aim is to find analytical formulae describing the outflow structure. In each binary system the more massive star is in the asymptotic giant branch and its wind is driven by a combination of pulsations in the stellar surface layers and radiation pressure on dust, while the less massive star is in the main sequence. Time averages of density and outflow velocity of the stellar wind are calculated and plotted as profiles against distance from the centre of mass and colatitude angle. We find that mass is lost mainly through the outer Lagrangian point L2. The resultant outflow develops into a spiral at low distances from the binary. The outflowing spiral is quickly smoothed out by shocks and becomes an excretion disk at larger distances. This leads to the formation of an outflow structure with an equatorial density excess, which is greater in binaries with smaller orbital separation. The pole-to-equator density ratio reaches a maximum value of $\sim10^5$ at Roche-Lobe Overflow state. We also find that the gas stream leaving L2 does not form a circumbinary ring for stellar mass ratios above 0.78, when radiation pressure on dust is taken into account. Analytical formulae are obtained by curve fitting the 2-dimensional, azimuthally averaged density and outflow velocity profiles. The formulae can be used in future studies to setup the initial outflow structure in hydrodynamic simulations of common-envelope evolution and formation of planetary nebulae., Comment: 13 pages, 17 figures, accepted for publication in MNRAS

Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefor, it usually pays off to compute local approximations of the used dynamical systems in the frequency and time domain. In this paper, we consider a structure-preserving extension of the frequency- and time-limited balanced truncation methods to second-order dynamical systems. We give a full overview about the first-order limited balanced truncation methods and extend those methods to second-order systems by using the different second-order balanced truncation formulas from the literature. Also, we present numerical methods for solving the arising large-scale sparse matrix equations and give numerical modifications to deal with the problematic case of second-order systems. The results are then illustrated on three numerical examples., Comment: 30 pages, 10 figures, 9 tables

We have studied the high-frequency dynamics of salol by inelastic x-ray scattering over a wide temperature range between 50 and 450 K, across the glass transition. We find that salol efficiently realizes the mechanism of dynamical arrest described by the mode-coupling theory, as manifested by a cusp singularity in the behaviour of the non-ergodicity parameter and a $Q$ dependence of the critical non-ergodicity parameter that is in phase with the static structure factor. These results confront positively the mode-coupling theory with liquids with local order., Comment: 7 pages, 6 figures

We present a model for the Galactic supernova remnant (SNR) VRO 42.05.01, suggesting that its intriguing morphology can be explained by a progenitor model of a supersonically moving, mass losing star. The mass outflows of the progenitor star were in the form of an asymmetric stellar wind focused on the equatorial plane of the star. The systemic motion of the parent star in combination with its asymmetric outflows excavated an extended wind bubble that revealed a similar structure to this of VRO 42.05.01. Currently the SNR is interacting with the wind bubble and it is dominantly shaped by it. Employing 2D hydrodynamic simulations we model VRO 42.05.01 under the framework of this model and we reproduce its overall morphological properties. We discuss the variations of our progenitor model in light of the current observational uncertainties., Comment: 5 pages, 3 figures

We present a polarimetric investigation of the protoplanetary nebula Frosty Leo performed with the Submillimeter Array. We were able to detect, in the low continuum level (peak at 14.4 mJy beam$^{-1}$), a marginal polarization at $\sim2.6\sigma$. The molecular line investigation based on the CO $J=3\rightarrow2$ emission shows a peak emission of 68.1 Jy beam$^{-1}$ km s$^{-1}$ and the polarization detection in this CO line is also marginal, with a peak at $\sim3.8\sigma$. In both cases, it was therefore not possible to use the electric vector maps (B-field) to accurately trace the magnetic field (B-field) within the PPN. The spatio-kinematic modelling realised with the different velocity channel maps indicates three main structures: a distorted torus accompanied by a bipolar outflow or jet aligned with its axis and a flattened spherical "cap". The comparison of the CO polarization segments with our model suggests that the polarized emission probably arises in the first two components., Comment: 8 pages, Accepted for publication in Monthly Notices of the Royal Astronomical Society

We present the results of long-slit echelle spectroscopy and deep narrow-band imaging of the Owl Nebula (NGC 3587), obtained at the \textit{Observatorio Astron\'omico Nacional, San Pedro M\'artir}. These data allow us to construct an iso-velocity data cube and develop a 3-D morpho-kinematic model. We find that, instead of the previously assumed bipolar dumbbell shape, the inner cavity consists of multi-polar fingers within an overall tripolar structure. We identify three additional planetary nebulae that show very similar morphologies and kinematics to the Owl, and propose that these constitute a new class of \textit{strigiform} (owl-like) nebulae. Common characteristics of the strigiform nebulae include a double-shell (thin outside thick) structure, low-luminosity and high-gravity central stars, the absence of a present-day stellar wind, and asymmetric inner cavities, visible in both optical and mid-infrared emission lines, that show no evidence for surrounding bright rims. The origin of the cavities is unclear, but they may constitute relics of an earlier stage of evolution when the stellar wind was active., Comment: 2018, MNRAS

Red novae are optical transients erupting at luminosities typically higher than those of classical novae. Their outbursts are believed to be caused by stellar mergers. We present millimeter/submillimeter-wave observations with ALMA and SMA of the three best known Galactic red novae, V4332 Sgr, V1309 Sco, and V838 Mon. The observations were taken 22, 8, and 14 yr after their respective eruptions and reveal the presence of molecular gas at excitation temperatures of 35-200 K. The gas displays molecular emission in rotational transitions with very broad lines (full width $\sim$400 km\s). We found emission of CO, SiO, SO, SO$_2$ (in all three red novae), H$_2$S (covered only in V838 Mon) and AlO (present in V4332 Sgr and V1309 Sco). No anomalies were found in the isotopic composition of the molecular material and the chemical (molecular) compositions of the three red novae appear similar to those of oxygen-rich envelopes of classical evolved stars (RSGs, AGBs, post-AGBs). The minimum masses of the molecular material that most likely was dispersed in the red-nova eruptions are 0.1, 0.01, and 10$^{-4}$ M$_{\odot}$ for V838 Mon, V4332 Sgr, and V1309 Sco, respectively. The molecular outflows in V4332 Sgr and V1309 Sco are spatially resolved and appear bipolar. The kinematic distances to V1309 Sco and V4332 Sgr are 2.1 and 4.2 kpc, respectively. The kinetic energy stored in the ejecta of the two older red-nova remnants of V838 Mon and V4332 Sgr is of order $10^{46}$ erg, similar to values found for some post-AGB (pre-PN) objects whose bipolar ejecta were also formed in a short-duration eruption. Our observations strengthen the link between these post-AGB objects and red novae and support the hypothesis that some of the post-AGB objects were formed in a common-envelope ejection event or its most catastrophic outcome, a merger., Comment: 19 pages, 13 figures, accepted to A&A

The Hankel-norm approximation is a model reduction method which provides the best approximation in the Hankel semi-norm. In this paper the computation of the optimal Hankel-norm approximation is generalized to the case of linear time-invariant continuous-time descriptor systems. An efficient algorithm is developed by refining the generalized balanced truncation square root method. For a wide practical usage, adaptations of the introduced algorithm towards stable computations and sparse systems are made as well as an approach for a projection-free algorithm. To show the approximation behavior of the introduced method, numerical examples are presented., Comment: 25 pages, 5 figures

Aims. The structural inhomogeneities and kinematics of massive star nebulae are tracers of their mass-loss history. We conduct a three-dimensional morpho-kinematic analysis of the ejecta of eta Car outside its famous Homunculus nebula. Methods. We carried out the first large-scale integral field unit observations of eta Car in the optical, covering a field of view of 1'x1' centered on the star. Observations with the Multi Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT) reveal the detailed three-dimensional structure of eta Car's outer ejecta. Morpho-kinematic modeling of these ejecta is conducted with the code SHAPE. Results. The largest coherent structure in eta Car's outer ejecta can be described as a bent cylinder with roughly the same symmetry axis as the Homunculus nebula. This large outer shell is interacting with the surrounding medium, creating soft X-ray emission. We establish the shape and extent of the ghost shell in front of the southern Homunculus lobe and confirm that the NN condensation can best be modeled as a bowshock in the orbital/equatorial plane. Conclusions. The SHAPE modeling of the MUSE observations indicates that the kinematics of the outer ejecta measured with MUSE can be described by a spatially coherent structure, and this structure also correlates with the extended soft X-ray emission associated with the outer debris field. The ghost shell just outside the southern Homunculus lobe hints at a sequence of eruptions within the time frame of the Great Eruption from 1837-1858 or possibly a later shock/reverse shock velocity separation. Our 3D morpho-kinematic modeling and the MUSE observations constitute an invaluable dataset to be confronted with future radiation-hydrodynamics simulations. Such a comparison may shed light on the yet elusive physical mechanism responsible for eta Car-like eruptions., Comment: 14 pages, 8 figures, accepted in A&A

Spatially resolved observations of the planetary nebula M2-42 (PN G008.2-04.8) obtained with the Wide Field Spectrograph on the Australian National University 2.3 m telescope have revealed the remarkable features of bipolar collimated jets emerging from its main structure. Velocity-resolved channel maps derived from the [N II] $\lambda$6584 emission line disentangle different morphological components of the nebula. This information is used to develop a three-dimensional morpho-kinematic model, which consists of an equatorial dense torus and a pair of asymmetric bipolar outflows. The expansion velocity of about 20 km s$^{-1}$ is measured from the spectrum integrated over the main shell. However, the deprojected velocities of the jets are found to be in the range of 80-160 km s$^{-1}$ with respect to the nebular center. It is found that the mean density of the collimated outflows, 595 $\pm$ 125 cm$^{-3}$, is five times lower than that of the main shell, 3150 cm$^{-3}$, whereas their singly ionized nitrogen and sulfur abundances are about three times higher than those determined from the dense shell. The results indicate that the features of the collimated jets are typical of fast, low-ionization emission regions., Comment: 6 pages, 4 figures, 4 tables, accepted for publication in The Astronomical Journal