Unstable Roots Research Articles

An analytical criterion for stability testing of delayed Bode’s transfer function with unity feedback

The unity feedback where delayed Bode’s transfer function is placed in forward path is used as reference transfer function in controller design for fractional time delay systems. In this study, bounded input-bounded output stability of this reference transfer function is investigated. This investigation is based on a theorem for fractional order time delay systems using Hassard’s techniques. Sufficient and necessary conditions are obtained for testing the stability of this reference transfer function. Moreover, an explicit formula is given for calculating the number of unstable roots in case of instability. The illustrative examples show that this stability criterion works efficiently.

Comparison of Functional Outcomes after Anterior Cruciate Ligament Reconstruction with Meniscal Repair for Unstable versus Stable Meniscal Tears.

This study aimed to compare functional outcomes including knee muscle strength in the quadriceps and hamstrings, and proprioception, assessed through dynamic postural stability (overall stability index [OSI]) and self-reported outcomes in the operated and non-operated knees between anterior cruciate ligament reconstruction (ACLR) with meniscal repair for unstable (root and radial tears) and stable (longitudinal, horizontal, and bucket handle tears) meniscal tears. A total of 76 patients were randomly selected (41 with ACLR with meniscal repair for unstable meniscal tears and 35 with ACLR with meniscal repair for stable meniscal tears) at three different time points (preoperative, 6 months, and 12 months). Repeated measures analysis of variance was used to investigate the differences in outcomes for between-subject and within-subject factors. In the operated knees, there were no significant differences for functional outcomes between the two groups (all p > 0.05). In the non-operated knees, a significant difference was observed for the OSI between the two groups, which was significantly higher in ACLR with meniscal repair for unstable meniscal tears than for stable meniscal tears at 6 months (p < 0.001). Multiple linear regression analysis showed that age (p = 0.027), preoperative OSI in the operated knees (p = 0.005), and postoperative OSI in the operated knees at 6 months (p = 0.002) were significant and independent predictors for OSI in the non-operated knees at 6 months postoperatively. Therefore, while no differences were observed in functional outcomes between the two groups in the operated knees, dynamic postural stability was poorer at 6 months postoperatively in the non-operated knees of patients with ACLR with meniscal repair for unstable meniscal tears. Furthermore, a significant correlation was observed between preoperative/postoperative dynamic postural stability in the operated knees and postoperative dynamic postural stability in the non-operated knees. Hence, we recommend incorporating balance exercises for both knees in post-surgical rehabilitation, particularly for patients with unstable meniscal tears.

Overview on Restoration with Endocrown

Endocrown restorations are single prostheses that are made of reinforced ceramics and may be acid etched; they are recommended for endodontically treated molar teeth that have suffered a considerable loss of coronal structure. In order to provide a strong foundation for the restoration and strengthen the restored tooth structurally, the quality and integrity of the residual tooth structure should be carefully preserved. For this reason, Endocrown is used as a minimally invasive technique. Endocrowns are particularly recommended for molars with short, destroyed, dilacerated, or unstable roots. They may also be utilized when it is unable to achieve an acceptable thickness of the ceramic coating on the metal or ceramic substructures due to significant coronal dental tissue loss and constrained interocclusal space. Endocrowns provide a more beneficial choice than conventional and post- and core-retained restorations because of the little and simple preparation required. This favorable outcome can be attributable to a number of variables, including the preparation method, occlusal thickness, and elastic moduli. When it comes to material selection it is impossible to create what is called the material of choice since new material is constantly being introduced.

Characterization of refill region and mixing state immediately ahead of a hydrogen-air rotating detonation using LES

The extraordinary complexity of real Rotating Detonation Combustors (RDC) demands a deep knowledge of each phenomenon involved in the wave development and propagation. Since the reactants are typically injected separately, a key element for the combustor design optimization is understanding the refilling process and the reactants mixing. However, due to the harsh environment and high working frequencies of RDCs, the experimental diagnostics is usually limited, so high-fidelity simulations represent an essential tool to complement the measurements with detailed insights into the flow. In the present work, the non-premixed RDC installed at TU Berlin is simulated with the AVBP code by solving the fully-compressible, spatially-filtered reactive Navier-Stokes equations. The complete hydrogen and air injection system is included in the numerical model to accurately describe both the reactants mixing and the complex turbulent flow field in the resulting refill region. This study shows how both the injection system configuration and its transient interaction with the wave are fundamental for the reactants mixing, as they directly influence the refilled gas properties. Limiting the imbalances between the blockage dynamics of the fuel and oxidizer ducts and optimizing the fuel injectors can improve considerably the homogeneity of the fresh mixture, and consequently the leading shock strength and reasonably the pressure gain. In fact, the detailed analysis of the detonation front speed shown a higher instability near the chamber base for the periodic presence of unmixed reactants. Nevertheless, the unstable root of the front does not affect the whole wave speed, and remaining part propagates steadily thanks to the tangential mixture uniformity. Moreover, the local speed distribution does not appear directly related to the small-scale mixture properties, indicating a higher sensibility to the annulus curvature rather than to the local gas state.

Nature and Causes of Economic Fluctuations: Evidence from the U.S. Based on a Nonlinear Autoregressive Integrated Process

Business cycles and economic growth have long been studied separately, hindering understanding of the nature and causes of economic fluctuations and growth. Here, we present an economic model that incorporates both deterministic trends and persistent fluctuations, derived from a general economic data generation process obtained using the dimensionless nonlinear autoregressive integrated (NLARI) process. To explore the nature and causes of economic fluctuations and growth, we develop a unified test to identify whether the economic data generation mechanism is stable fixed points, unit roots, cyclical oscillations, or chaos. Empirical studies show that as the economic resilience coefficient changes from strong to weak, the data-generating mechanism for U.S. economic indicators shifts from a stable fixed point (dynamic equilibrium) in the supply-side economy to an unstable unit root (divergence) in the demand-side economy involving income distribution. Therefore, the U.S. economy is in partial dynamic equilibrium of the supply-side economy. The low resilience of the demand-side economy allows for the accumulation of noise from exogenous shocks, leading to persistent fluctuations. We find that the economic deterministic trend is ([Formula: see text]/[Formula: see text])[Formula: see text], i.e., time [Formula: see text] multiplied by the ratio of the mean of exogenous supply, demand, and technological progress investment shocks to the economic resistance coefficient. A general dynamic equilibrium economy maximizes growth from investment by avoiding [Formula: see text] growth losses from permanently persistent fluctuations. Thus, ([Formula: see text]/[Formula: see text])[Formula: see text] is the maximum value obtained along the maximum profit growth at the von Neumann equilibrium price. The investment path that creates the maximum ratio [Formula: see text]/[Formula: see text] is the optimal growth path.

Stability and bifurcations in scalar differential equations with a general distributed delay

For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model.

Three-dimensional coupling of electron cyclotron drift instability and ion–ion two stream instability

Electron cyclotron drift instability (ECDI) and ion–ion two stream instability (IITSI) are both kinetic instabilities that can be present in low-temperature, partially magnetized plasmas. The coupling of instabilities in a three-dimensional configuration leads to the existence of more than one unstable roots to the kinetic dispersion relation. In this paper, a generalized method has been developed for numerically evaluating solutions to the three-dimensional dispersion relation for coupled ECDI and IITSI, assuming cold singly and doubly charged ions and a Maxwellian velocity distribution function for the electrons. The present study demonstrates the coupling between ECDI and IITSI that affects the most unstable mode as a function of the wavenumbers in three dimensions and various plasma properties, including the applied electric field, magnetic field, electron temperature, ion velocities, and plasma density. One of the most notable results is that, while the most unstable mode with the largest growth rate is in the direction of the E×B drift in the two-dimensional cases, the most unstable mode for the three-dimensional configuration occurs in the oblique direction between the applied electric field and the E×B drift. This agrees with experimental observations in cross field plasma sources using coherent Thomson scattering.

A Sufficient Statistical Test for Dynamic Stability

In the existing Statistics and Econometrics literature, there does not exist a statistical test which may test for all kinds of roots of the characteristic polynomial leading to an unstable dynamic response, i.e., positive and negative real unit roots, complex unit roots and the roots lying inside the unit circle. This paper develops a test which is sufficient to prove dynamic stability (in the context of roots of the characteristic polynomial) of a univariate as well as a multivariate time series without having a structural break. It covers all roots (positive and negative real unit roots, complex unit roots and the roots inside the unit circle whether single or multiple) which may lead to an unstable dynamic response. Furthermore, it also indicates the number of roots causing instability in the time series. The test is much simpler in its application as compared to the existing tests as the series is strictly stationary under the null (C01, C12).

Growth patterns and environmental adaptions of the tree species planted for ecological remediation in typhoon-disturbed areas—A case study in Zhuhai, China

Typhoon frequently results in various mechanical damages to urban forest ecosystems. Imperative forest remediation projects were launched to restore the environmental conditions in cities, in which massive trees were newly planted. However, it was rarely answered whether the newly planted trees could acclimate to typhoon circumstances and enhance the wind resistance of the local ecosystem. Therefore, it was necessary to achieve information on the physical growth and windy environmental adaption of newly planted trees, which could promote a profound understanding of the efficiency of post-typhoon ecological remediation. In this study, we selected Zhuhai's urban-forest remediation district as our research area that suffered severely from Typhoon Hato (2017). The six newly-planted tree species for the ecological remediation were measured for their above- and below-ground processes from June 2018 to December 2019, including their development of tree height, ground diameter, crown size, and fine root biomass. Additionally, the variations of the soil's physical and chemical properties were also measured to assess the impact of plantation on soil conditions. Our results showed that the six surveyed tree species had different above- and below-ground growth patterns. With robust root development at horizontal and vertical levels combined with relatively short and thick above-ground profiles, Sterculia lanceolata Cav. and Cinnamomum camphora (Linn) were likely to cope well with typhoon disturbances. Ilex rotunda Thunb. and Schima superba Gardn. et Champ. exhibited moderate acclimation to windy environment, while Elaeocarpus sylvestris (Lour.) Poir. and Elaeocarpus apiculatus Mast. were not recommended to be planted in typhoon-disturbed areas concerning their unstable root development. In addition, the ecological remediation did improve the soil properties, specifically for the chemical characteristics including available nitrogen, available potassium, and soil organic matter. To improve the effectiveness of forest remediation in the future, it was better to choose those tree species with vigorous root development and steady values of root:shoot ratios, which might be advantageous for coping with typhoon disturbances. The tree species with prosperous above-ground growth were not suitable for areas facing strong winds directly but could be planted in leeward regions to amplify their landscape functions.

An algorithm to design a stabilizing feedback controller for linear delay systems via Parseval ’s theorem

We investigate feedback stabilization problems of a linear delay system. When the unstable characteristic roots of the system are far from the imaginary axis, the discretization of the system results in a large error. In this case, it is difficult to seek stabilizing control laws via the algorithm by Hu and Hu, (2020). To avoid the discretization of unstable differential equations, a modified state equation is constructed through a shifting parameter such that the equation is asymptotically stable. Then, based on the modified state equation and Parseval’s theorem, a numerical optimization algorithm is provided to design a stabilizing controller. Meanwhile, we compare the presented algorithm with those in the literature. Finally, numerical examples show that the presented algorithm in this paper is efficient.

The other side of tropical forest drought: do shallow water table regions of Amazonia act as large-scale hydrological refugia from drought?

Tropical forest function is of global significance to climate change responses, and critically determined by water availability patterns. Groundwater is tightly related to soil water through the water table depth (WT), but historically neglected in ecological studies. Shallow WT forests (WT < 5 m) are underrepresented in forest research networks and absent in eddy flux measurements, although they represent c. 50% of the Amazon and are expected to respond differently to global-change-related droughts. We review WT patterns and consequences for plants, emerging results, and advance a conceptual model integrating environment and trait distributions to predict climate change effects. Shallow WT forests have a distinct species composition, with more resource-acquisitive and hydrologically vulnerable trees, shorter canopies and lower biomass than deep WT forests. During 'normal' climatic years, shallow WT forests have higher mortality and lower productivity than deep WT forests, but during moderate droughts mortality is buffered and productivity increases. However, during severe drought, shallow WT forests may be more sensitive due to shallow roots and drought-intolerant traits. Our evidence supports the hypothesis of neglected shallow WT forests being resilient to moderate drought, challenging the prevailing view of widespread negative effects of climate change on Amazonian forests that ignores WT gradients, but predicts they could collapse under very strong droughts.

A characteristic MRI finding to diagnose a partial tear of the medial meniscus posterior root: an ocarina sign

BackgroundDiagnosing partial tears of the medial meniscus (MM) posterior root is difficult. The aim of this study was to evaluate diagnostic values involved in conventional magnetic resonance imaging (MRI) features of MM posterior root tears (MMPRTs) and find other MRI-based findings in patients with partial MMPRTs.MethodsEighteen patients who had arthroscopically confirmed partial MMPRTs were included. As a control, 18 patients who underwent partial meniscectomy for other types of MM tears were evaluated. Isolated partial MMPRTs were classified into the following three types: type A, accurate partial stable tear (cleavage < 1/2 of root width); type B, bridged unstable root tear (cleavage ≥ 1/2 of root width); type C, complex horn tear expanded to the root. Conventional MRI-based findings of MMPRTs were evaluated between two groups (n = 23). Posterior root irregularity, bone marrow spot, and ocarina-like appearance showing several condensed circles in triangular meniscal horn (ocarina sign) were also evaluated.ResultsPosterior root irregularity and bone marrow spot were frequently observed in the partial MMPRTs (47.8%), compared with the other MM tears (P = 0.007 and 0.023, respectively). The ocarina sign was detected in 69.6% of patients with partial MMPRTs. A significant difference between two groups was observed in a positive ratio of ocarina sign (P < 0.001). Types A, B, and C of the partial tear/damage were observed in three, eight, and seven patients, respectively. The ocarina sign was the most common MRI finding in each type of partial MMPRT.ConclusionsThis study demonstrated that a characteristic MRI finding, “ocarina sign,” was frequently observed in patients with partial tear/damage of the MM posterior root. The ocarina sign was the most common MRI finding in several types of partial MMPRTs. Our results suggest that the ocarina sign may be useful to diagnose unnoticed partial MMPRTs.Level of evidence: IV, retrospective comparative study.

Dynamical invariants and inverse period-doubling cascades in multi-delay systems.

We investigate transitions to simple dynamics in first-order nonlinear differential equations with multiple delays. With a proper choice of parameters, a single delay can destabilize a fixed point. In contrast, multiple delays can both destabilize fixed points and promote high-dimensional chaos but also induce stabilization onto simpler dynamics. We show that the dynamics of these systems depend on the precise distribution of the delays. Narrow spacing between individual delays induces chaotic behavior, while a lower density of delays enables stable periodic or fixed point behavior. As the dynamics become simpler, the number of unstable roots of the characteristic equation around the fixed point decreases. In fact, the behavior of these roots exhibits an astonishing parallel with that of the Lyapunov exponents and the Kolmogorov-Sinai entropy for these multi-delay systems. A theoretical analysis shows how these roots move back toward stability as the number of delays increases. Our results are based on numerical determination of the Lyapunov spectrum for these multi-delay systems as well as on permutation entropy computations. Finally, we report how complexity reduction upon adding more delays can occur through an inverse period-doubling sequence.

Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

Supermatrix phylogeny resolves goby lineages and reveals unstable root of Gobiaria

Gobies, sleepers, and cardinalfishes represent major clades of a species rich radiation of small bodied, ecologically diverse percomorphs (Gobiaria). Molecular phylogenetics has been crucial to resolving broad relationships of sleepers and gobies (Gobioidei), but the phylogenetic placements of cardinalfishes and nurseryfishes, as reciprocal or sequential sister clades to Gobioidei, are uncertain. In order to evaluate relationships among and within families we used a phylogenetic data mining approach to generate densely sampled trees inclusive of all higher taxa. We utilized conspecific amino acid homology to improve alignment accuracy, included ambiguously identified taxa to increase taxon sampling density, and resampled individual gene alignments to filter rogue sequences before concatenation. This approach yielded the most comprehensive tree yet of Gobiaria, inferred from a sparse (17 percent-complete) supermatrix of one ribosomal and 22 protein coding loci (18,065 characters), comprised of 50 outgroup and 777 ingroup taxa, representing 32 percent of species and 68 percent of genera. Our analyses confirmed the lineage-based classification of gobies with strong support, identified sleeper clades with unforeseen levels of systematic uncertainty, and quantified competing phylogenetic signals that confound resolution of the root topology. We also discovered that multilocus data completeness was related to maximum likelihood branch support, and verified that the phylogenetic uncertainty of shallow relationships observed within goby lineages could largely be explained by supermatrix sparseness. These results demonstrate the potential and limits of publicly available sequence data for producing densely-sampled phylogenetic trees of exceptionally biodiverse groups.

On Topological Properties of the Set of Stabilizing Feedback Gains

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous and discrete time linear time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states have at most [n/2] connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LTI systems; the proposed algorithm also computes the number of unstable roots in each unstable interval.

A reliable graphical criterion for TDS stability analysis

The paper considers the stability issue of linear systems with commensurate delays. This issue can be well characterised by the distribution of roots of system's characteristic equation. At first, distribution boundary of the roots (with positive real parts) is explicit given in a practical way. Subsequently, a reliable graphical stability criterion for calculating the number of unstable roots is deduced, associating with auxiliary polynomial which plays an important role in the analysis of high order and complex systems. Moreover, a procedure for drawing the winding curve of characteristic function in finite path is proposed. At last, typical examples are given to illustrate that the result carried out is reliable and efficient.

A Current Sensorless Delay–Based Control Scheme for MPPT–Boost Converters in Photovoltaic Systems

The results presented in this paper deal with the design of a current sensorless delay–based controller for the closed–loop stabilization of a photovoltaic system under an MPPT scheme using a boost dc/dc converter. Some applications of such topology are dc microgrids, solar vehicles, or stand-alone systems, to mention a few. The basis of this control scheme relies on the feedback linearization control technique coupled with a delay–based low-order controller. In order to study the stability, the proposed approach uses a geometric point of view which allows the partitioning of the controller parameters space into regions with similar stability characteristics (same number of unstable characteristic roots). The most important contribution of the paper relies on providing practical guidelines to tune the gains of the proposed delay–based controller, ensuring asymptotic stability of the closed–loop system and fulfilling the requirements for photovoltaic applications. In addition, the proposed approach allows the design a non-fragile controller with respect to the controller gains. Furthermore, in order to test the effectiveness of the control scheme presented, experimental results evaluating the closed–loop system performance under set-point changes and abrupt irradiance disturbances are addressed using a solar array simulator and a battery bank as load.

A Sturm–Liouville theorem for quadratic operator pencils

We establish a Sturm–Liouville theorem for quadratic operator pencils with matrix-valued potentials counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

Stability criteria for neutral delay differential-algebraic equations with many delays

In this paper the asymptotic stability is concerned for a class of neutral delay differential-algebraic equations (NDDAEs). We will present two criteria by evaluating a corresponding harmonic function on the boundary of a torus region. Stability regions are also presented so as to locate all possible unstable characteristic roots of NDDAEs. As we know, these roots will make bad numerical simulations. Our criteria help find and avoid them. Numerical examples are shown to check our criteria.