Invariant Approximation Research Articles

Mizoguchi-Takahashi local contractions to Feng-Liu contractions

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.

Solving a class of infinite‐dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

AbstractWe examine a method for solving an infinite‐dimensional tensor eigenvalue problem , where the infinite‐dimensional symmetric matrix exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to is used to obtain an approximation to the desired eigenvector. This infinite‐dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step to ensure accurate and rapid convergence of the power method.

Invariant approximation in 2-banach space with $$H^{+}$$ mappings

Invariant approximation in 2-banach space with $$H^{+}$$ mappings

The fixed point theorems and invariant approximations for random nonexpansive mappings in random normed modules

By making full use of the theory of random sequential compactness in random normed modules, in this paper we establish a noncompact Dotson fixed point theorem: if C is a σ–stable random sequentially compact L0–star–shaped subset of a random normed module, then every random nonexpansive mapping T:C→C has a fixed point. Furthermore, we obtain an existence result for best approximations in random normed modules: let E be a random normed module, T:E→E a random nonexpansive mapping with a fixed point u and C a closed, σ–stable and T–invariant subset of E such that T(C)‾ is random sequentially compact, then the set of best approximations of u in C is nonempty, which generalizes the classical result of Smoluk. In addition, we also get an existence result for invariant approximations in random normed modules. A significant distinction between the proofs of our results in random normed modules and the corresponding classical results in normed spaces is that the σ–stability of both the sets and mappings involved in the random setting plays a prominent part in the proofs of the main results of this paper.

Statistical analysis of hypersonic glide vehicle radar cross section

AbstractThe capability to design, manufacture and test Hypersonic Glide Vehicles (HGVs) has been demonstrated by a number of nations, and they are increasingly forming part of military inventories, potentially offering capabilities highly unique to this technology. This article reports the simulated Monostatic Radar Cross Section of a generic HGV in five frequency ranges, HF, VHF, UHF, L, and S‐bands associated with different radar types. Full spherical datasets of complex co‐ and cross‐polar data are synthesised so that backscatter resulting from illumination by r.f./microwave energy of linear or circular polarisation can subsequently be computed from the raw dataset. Circular polarisation is commonly employed by ground‐based Ballistic Missile Early Warning Systems and Space Object Surveillance and Identification radars to avoid polarisation mis‐match losses resulting from ionospheric Faraday rotation effects. The data was generated using Ansys' Finite Element Solver at 10, 150 and 430 MHz, with the Geometric Optics/Physical Optics based SBR+ solver employed for 1.3 and 3 GHz data. All data was produced at below the Nyquist sampling interval relevant to the target's electrical size. These datasets were then imported into a Matlab routine which extracted data over limited angular ranges associated with the likely radar line‐of‐sight in particular scenarios, typically having a standard deviation of ±10° about the direction of flight, applying either a Gaussian or Uniform sampling distribution as part of a Monte Carlo analysis. These extracted data were then used to form histograms giving the probability of sampling particular RCS values. Probability density functions and cumulative distribution functions were then fitted, to aid in the representation of statistical target fluctuations for each band and angular sampling range. The HGV exists in either the ‘Rayleigh’, ‘resonance’ or ‘optical’ scattering regimes, depending on its relative electrical size. The results suggest that for this target shape at HF and VHF cases a simple Swerling 0 (fluctuation invariant) approximation is adequate in most instances, whilst a Gamma distribution may be applied for UHF band cases. At L and S‐band a Beta distribution was found to provide a good fit to the available data.

Electric dipole response in 156–170Er nuclei

For the first time, this study investigated an energy region that covers both the Pygmy Dipole Resonance (PDR) and the Giant Dipole Resonance (GDR) for a broad set of even-even erbium isotopes. More specifically, translational and Galilean invariant (TGI-) quasiparticle random phase approximation (QRPA) was applied to investigate eight even-even 156–170Er isotopes. Through cross-section calculations, we obtained information about the structure of the GDR and PDR modes, as well as the splitting between the K = 0 and K = 1 branches. The PDR region was explored in more detail by investigating the region up to both 8 MeV and 10 MeV to determine the main distribution area of the PDR. It was found that only the E1 strengths below 8 MeV matched well with the well-known systematics of the PDR.

Ellipsoidal approximations of the minimal robust positively invariant set

An optimization algorithm is presented in this paper for the minimal robust positively invariant (mRPI) set approximations via sums-of-squares (SOS) optimization. The mRPI set is an effective tool for robust analysis of uncertain systems under bounded disturbances. The approximation of the mRPI set is always characterized by a polyhedron computed after finite time iterations. In this paper, an mRPI set is characterized by an ellipsoidal set while bounded parametric uncertainties act on states. The proposed algorithm optimizes the shape matrix of the ellipsoidal set approximation by minimizing the volume of the ellipsoidal set. The algorithm is designed for discrete-time and continuous-time nonlinear systems respectively. The algorithm has the ability to further minimize the mRPI set by optimizing the state-feedback control law. Examples are employed to validate the effectiveness of the proposed algorithms.

Computation of invariant sets via immersion for discrete-time nonlinear systems

In this paper, we propose an approach for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we focus on the maximal admissible invariant set contained in some given constraint set. For special types of nonlinear systems, which can be exactly immersed into higher dimensional linear systems with state transformations, invariant sets of the original nonlinear system can be characterized using the higher dimensional linear representation. For general nonlinear systems without the immersibility property, approximate immersions are defined in a local region within some tolerance and linear approximations are computed by leveraging the fixed-point iteration technique for invariant sets. Given the bound on the mismatch between the linear approximation and the original system, we provide an invariant inner approximation of the maximal admissible invariant set by a tightening procedure.

Orbital Flips Caused by the Eccentric Von Zeipel–Lidov–Kozai Effect in Nonrestricted Hierarchical Planetary Systems

The eccentric von Zeipel–Lidov–Kozai effect is widely applied to diverse astrophysical settings. In the restricted hierarchical three-body system, the topic of orbital flips has been extensively studied. However, it is far from being understood for nonrestricted circumstances. In this work, the dynamics of orbital flips are investigated under the Hamiltonian model at the octupole-level approximation for nonrestricted hierarchical planetary systems, where the outer planet is more massive than the inner one. Numerical distribution of flipping orbits shows that there are three major flipping regions, distributed in low-, intermediate-, and high-eccentricity spaces. Poincaré sections indicate that those islands of libration centered near i tot = 90° can lead to orbital flips. Thus, we refer to the behavior of orbital flips as a resonance phenomenon. From this viewpoint, dynamical models of orbital flips can be described by a separable Hamiltonian, which can be treated by a means of perturbation theory. The resonant model for orbital flips is formulated based on the adiabatic invariant approximation and then phase portraits are generated by plotting level curves of adiabatic invariants with the given Hamiltonian. By analyzing phase portraits, analytical boundaries of libration and circulation zones causing orbital flips are obtained. As expected, the numerical and analytical conditions that allow orbits to flip agree well with each other. The phenomenon of orbital flips in nonrestricted hierarchical problems can be well understood with the help of dynamical structures of secular resonance.

Some Results on Invariant Best Approximations in p-Normed Spaces

In this paper we give two theorems in p-normed spaces. The first one is about the existence of fixed point of multivalued nonexpansive mapping and then we use these theorems in the proof of the result( Theorem2.3 )about existence of invariant best approximation. Note that, we need to give Lemma (1.14)

The Von Zeipel–Lidov–Kozai Effect inside Mean Motion Resonances with Applications to Trans-Neptunian Objects

Secular dynamics inside mean motion resonances (MMRs) plays an essential role in governing the dynamical structure of the trans-Neptunian region and sculpting the orbital distribution of trans-Neptunian objects (TNOs). In this study, semianalytical developments are made to explore the von Zeipel–Lidov–Kozai resonance inside MMRs. To this end, a semi-secular model is formulated from averaging theory and then a single-degree-of-freedom integrable model is achieved based on the adiabatic invariance approximation. In particular, we introduce a modified adiabatic invariant, which is continuous around the separatrices of MMRs. During long-term evolution, both the resonant Hamiltonian and the adiabatic invariant remain unchanged, thus phase portraits can be produced by plotting level curves of the adiabatic invariant with a given Hamiltonian. The phase portraits provide global pictures to predict long-term behaviors of the eccentricity, inclination, and argument of pericenter. Applications to some representative TNOs inside MMRs (2018 VO137, 2005 SD278, 2015 PD312, Pluto, 2004 HA79, 1996 TR66, and 2014 SR373) show good agreements between the numerically propagated trajectories under the full N-body model and the level curves arising in phase portraits. Interestingly, 2018 VO137 and 2005 SD278 exhibit switching behaviors during their long-term evolution and currently they are inside 2:5 MMR with Neptune.

Role of the quadrupole deformation in γ-soft nuclei: The case of 124−134Xe

Observation of the scissors mode in the γ-soft and transition nuclei provides the reasoning for investigating the possibility of quadrupole deformation in such nuclei. Using a deformed-based model for an investigation of the scissors mode, the goal of this work was to analyze the effects associated with quadrupole deformation for the so-called “γ-soft” nuclei in the example of even-even 124−134Xe isotopes. Here, the well-known properties of the scissors mode were investigated using just the quadrupole deformed-based rotational, invariant quasiparticle random phase approximation (QRPA) method. The properties of the scissors mode were clearly seen for all the investigated xenon isotopes, demonstrating that quadrupole deformed-based models can easily explain the nature of the low-lying M1 strength for γ-soft nuclei and indicating that quadrupole deformation plays a non-negligible role in the formation of the low-lying dipole strength for such nuclei.

Invariant Domain-Preserving Approximations for the Euler Equations with Tabulated Equation of State

Invariant Domain-Preserving Approximations for the Euler Equations with Tabulated Equation of State

Bistatic SAR Point Spread Function Analysis for Close Proximity Geometries

The goal of this project is to describe the spatially variant imaging performance of bistatic SAR imaging using a localized spatially invariant approximation so as to use spatial Fourier analysis. Bistatic synthetic aperture radar (SAR) has been proposed for use in search and rescue operations, for UAV guidance, and as an all-weather navigational aid for landing aircraft. These applications involve aggressive maneuvering platforms operating at short standoff ranges. Matched filter image formation (MFIF) and the equivalent back-projection image formation algorithm fully capture the intrinsic imaging performance of these geometries. The imaging performance for these imaging configurations is described by the spatially variant point spread function (PSF). The SAR imaging process is described as a non-convolutional integral operation between the scene’s reflectivity function and the PSF. It is desirable to describe the process as a multiplication operation in the spatial Fourier domain but this requires the imaging process to be spatially invariant. The model derived here deals with the spatial variations by segmenting a large scene into smaller regions. Within each region, the spatially variant PSF is locally approximated as a spatially invariant impulse response (IPR). The SAR imaging process is described as a localized convolution operation. This approximation is extremely accurate at the center of a segmented region but degrades for positions displaced from the center. Error analysis is used to determine the maximum size of a segment to limit the error to a given tolerance.

On Invariant Approximations in Modular Spaces

This article is devoted to presenting results on invariant approximations over a non-star-shsped weakly compact subset of a complete modular space by introduced a new notion called S-star-shaped with center f: if be a mapping and , . Then the existence of common invariant best approximation is proved for Banach operator pair of mappings by combined the hypotheses with Opial’s condition or demi-closeness condition

Electric dipole strength in the deformed 144,146,148,150,152,154Nd nuclei

The Giant Dipole and Pygmy Dipole strengths distributions in the chain of the double-even 144−154Nd nuclei were investigated using a translational and Galilean invariant quasiparticle random phase approximation (TGI-QRPA). Through the cross sections, transition probabilities and other properties, such as the splitting behaviors of the electric dipole response between the K=0 and K=±1 components of the electric dipole response were investigated. The predicted concentration of the Giant Dipole Resonance (GDR) exhausts than 99% of the energy weighed sum rules (EWSR) below 20 MeV. Where up to the energy 8 MeV the highly-fragmented E1 strength was predicted. The calculations revealed that neutron excess together with deformation significantly increases the total low-energy E1 strength, while when increasing the deformation, K=0 fills the Pygmy Dipole Resonance (PDR) region more strongly than K=±1. Here also through nucleon structures such as collectivity and isospin structure of the excitations were presented discussion on differences between PDR and GDR region for neodymium isotopes. In addition, our results reveal that the spin-flip M1 strengths exhibit a broad distribution between 5 and 8 MeV and occupy less than 8% of the total dipole strength in that energy region.

Enhanced spin–orbit coupling in the underscreened Anderson lattice model for itinerant 5f metals

We investigate the effect of many-body interactions on the spin–orbit coupling of anisotropic metals. We use the underscreened Anderson lattice model that was proposed to describe uranium and plutonium compounds. By using a rotationally invariant approximation, we show that Coulomb interactions induce off-diagonal correlations that enhance the components of the spin–orbit coupling. Even modest values of the Coulomb interaction U can enhance the spin–orbit coupling by factors of about 4 and have significant effects on the electronic spectrum, but the enhancements are most pronounced for systems that are on the verge of magnetic instabilities. The enhancement is anisotropic in crystals with non-cubic symmetries and can lead to giant magnetic anisotropies in paramagnetic states.

Common Fixed Point and invariant Approximations for Cq -Commuting Mappings in P-Normed Spaces

The purpose of this paper is to prove a common fixed point(c.f.p) theorems by using conditiond (S(x), T(y)) ≤ℓ max{d(h(x), G(y)), d(h(x), S(x)), d(G(y), T(y)), d(h(x), T(y)), d(G(y), S(x))} For two pairs of mappings in p-normed space(p-n.s) and also obtain the best approximation (b.a) result. In the last part of this paper, it is proved that the fixed point (f.p) problem for these mappings is well-posed (w-p).

IR spectroscopy of condensed phase systems: Can the environment induce vibrational mode coupling?

The Perturbed Matrix Method is one of the methods to carry out theoretical studies on infrared spectroscopy at a low computational cost. One of the main assumptions adopted up to now is the invariant mode approximation: the actual mass-weighted Hessian eigenvectors of a vibrational center embedded in a chemical environment can be well approximated by the vacuum Hessian eigenvectors with the environment perturbation affecting only the mode frequencies. Herein, we investigate the accuracy of this approximation considering different scenarios. The results demonstrate the accuracy of this approximation providing non-negligible mode coupling only when dealing with extremely intense perturbation fields.

Pygmy dipole resonance in the well-deformed even–even 152–162Gd

The electric dipole E1 strength in the well-deformed even–even 152–162Gd was calculated using a model based on translational and Galilean invariant quasi-particle random phase approximation (QRPA), where for the first time the highly-fragmented E1 strength has been predicted below the particle threshold energy. The predicted concentration of the electric dipole excitations exhausts less than 2% of the energy weighed sum rules (EWSR) below 20 MeV in rather good agreement with the experimental data for neighbor nuclei. Through cross sections, transition probabilities, and other properties, the electric dipole response was investigated in 152–162Gd for a wide range of excitation energies from 5 to 20 MeV. The calculation that was performed with the elimination of effects associated with deformation revealed that the effects associated with neutron excess play a more significant role in the formation of the pygmy dipole resonance (PDR) strength when compared to the effects associated with a changing deformation. The calculations indicate the presence of magnetic dipole excitations with summed M1 strengths in the energy region between 5 and 8 MeV was less than 10% of the total dipole strength in the same energy region. We also took into account a recent experimental study of 156Gd in our discussion and paid particular attention to this isotope.