Invertibility Properties Research Articles

The invertible electrochemical properties and thermal response of a series of gel-type ionic liquids based on polyoxometalates.

A series of vanadium-substituted Dawson-structure POM-type ionic liquids, [TEAPS]7P2W17VO62 and [TEAPS]9P2W15V3O62, bearing sulfo-group grafted ammonium (TEAPS) cations and Dawson-type polyoxoanions have been formed which are reversible-thermal-response type gels. These gel-type compounds exhibit a phase transition from a quasi-solid gel phase to an isotropic sol phase. What's more, this series of hybrid compounds can undergo reversible electrochemical reactions in dimethyl formamide (DMF) owing to the reduction of the vanadium in POM anions as a simple anion, which is unlikely to happen in water solution because of water protonation.

A symplectic extension map and a new Shubin class of pseudo-differential operators

For an arbitrary pseudo-differential operator A:S(Rn)→S′(Rn) with Weyl symbol a∈S′(R2n), we consider the pseudo-differential operators A˜:S(Rn+k)→S′(Rn+k) associated with the Weyl symbols a˜=(a⊗12k)∘s, where 12k(x)=1 for all x∈R2k and s is a linear symplectomorphism of R2(n+k). We call the operators A˜ symplectic dimensional extensions of A. In this paper we study the relation between A and A˜ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of A˜ in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes HGρm1,m0 of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HGρm1,m0 but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.

Image reconstruction from a complete set of geometric and complex moments

An image can be reconstructed from the finite set of its orthogonal moments. Since geometric and complex moment kernels do not satisfy orthogonality criterion, direct image reconstruction using them is deemed to be difficult. In this paper, we propose a technique to reconstruct an image from either geometric moments (GMs) or complex moments (CMs). We utilize a relationship between GMs and Stirling numbers of the second kind. Then, by using the invertibility property of the Stirling transform, the original image can be reconstructed from its complete set of either geometric or complex moments. Further, based on previous works on blur effects on a moment domain and using the proposed reconstruction methods, a formulation is shown to obtain an estimated original image from the degraded image moments and the blur parameter. The reconstruction performance of the proposed methods on blur images is presented to validate the theoretical framework.

Dilation of the Weyl symbol and Balian-Low theorem

The key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their Weyl symbols. This observation has two implications in time-frequency analysis. First, it implies the stability of general Gabor frames under small dilations of the time-frequency set, previously known only for the case that the time-frequency set is a lattice. Secondly, it allows us to derive a new Balian-Low theorem (BLT) for Gabor systems with window in the standard window class and with general time-frequency families. In contrast to the classical versions of BLT the new BLT does not only exclude orthonormal bases and Riesz bases at critical density, but indeed it even excludes irregular Gabor frames at critical density.

A New VSC-HVDC Model for Power Flows Using the Newton-Raphson Method

The paper presents a new model of the VSC-HVDC aimed at power flow solutions using the Newton-Raphson method. Each converter station is made up of the series connection of a voltage source converter (VSC) and its connecting transformer which is assumed to be a tap-changing (LTC) transformer. The new model represents a paradigm shift in the way the fundamental frequency, positive sequence modeling of VSC-HVDC links are represented, where the VSCs are not treated as idealized, controllable voltage sources but rather as compound transformer devices to which certain control properties of PWM-based inverters may be linked—just as DC-to-DC converters have been linked, conceptually speaking, to step-up and step-down transformers. The VSC model, and by extension that of the VSC-HVDC, takes into account, in aggregated form, the phase-shifting and scaling nature of the PWM control. It also takes into account the VSC inductive and capacitive reactive power design limits, switching losses and ohmic losses.

Perturbations of Invertible Operators and Stability of g-Frames in Hilbert Spaces

In this paper, we study the perturbations of invertible operators and stability of g-frames in Hilbert spaces. In particular, we obtain some conditions under which the perturbations of an invertible operator are still an invertible operator, the perturbations of a right invertible operator or a surjective operator are still a right invertible operator or surjective operator. Then we apply the perturbations of invertible operators to study the stability of g-frames which is close related with the invertibility (or right invertibility) property of operators.

Dirichlet‐transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds

AbstractIn this paper we use a layer potential analysis to show the existence of solutions for a Dirichlet‐transmission problem for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in compact boundaryless Riemannian manifolds. Compactness and invertibility properties of corresponding layer potential operators on Lp, Sobolev, or Besov scales are also obtained.

Diabatic processes modifying potential vorticity in a North Atlantic cyclone

AbstractPotential vorticity (PV) succinctly describes the evolution of large‐scale atmospheric flow because of its material conservation and invertibility properties. However, diabatic processes in extratropical cyclones can modify PV and influence both mesoscale weather and the evolution of the synoptic‐scale wave pattern. In this investigation, modification of PV by diabatic processes is diagnosed in a Met Office Unified Model (MetUM) simulation of a North Atlantic cyclone using a set of PV tracers. The structure of diabatic PV within the extratropical cyclone is investigated and linked to the processes responsible for it. On the mesoscale, a tripole of diabatic PV is generated across the tropopause fold extending down to the cold front. The structure results from a dipole in heating across the frontal interface due to condensation in the warm conveyor belt flanking the upper side of the fold and evaporation of precipitation in the dry intrusion and below. On isentropic surfaces intersecting the tropopause, positive diabatic PV is generated on the stratospheric side, while negative diabatic PV is generated on the tropospheric side. The stratospheric diabatic PV is generated primarily by long‐wave cooling which peaks at the tropopause itself due to the sharp gradient in humidity there. The tropospheric diabatic PV originates locally from the long‐wave radiation and non‐locally by advection out of the top of heating associated with the large‐scale cloud, convection and boundary layer schemes. In most locations there is no diabatic modification of PV at the tropopause itself but diabatic PV anomalies would influence the tropopause indirectly through the winds they induce and subsequent advection. The consequences of this diabatic PV dipole for the evolution of synoptic‐scale wave patterns are discussed.

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab = ba, we show that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a + b) D in terms of a, a D , b and b D , but also (1 + a D b) D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.

Factors determining the catalytic activity of the mixed micellar system cetyltrimethylammonium bromide-Brij-35 in the hydrolysis of a phosphonic acid ester

The micellization and surface properties, Stern layer micropolarity, and catalytic action of the cetyltrimethylammonium bromide-Brij-35 system in the alkaline hydrolysis of O-ethyl O-p-nitrophenyl chloromethylphosphonate have been investigated at different proportions of the constituent surfactants. This system has unique, invertible catalytic properties ranging from acceleration of the hydrolysis reaction by more than one order of magnitude to strong inhibition of this reaction and even termination of the reaction at certain proportions of the surfactants. The factors in the changes of the catalytic effect under variation of the surfactant ratio are micropolarity in the reaction zone and surface potential. A likely cause of the termination of the process is the shift of the reaction zone from the surface layer to the micelle core, where the environment is less polar.

Micellar assemblies from amphiphilic polyurethanes for size-controlled synthesis of silver nanoparticles dispersible both in polar and nonpolar media

A size-controlled synthesis of silver nanoparticles (AgNPs) dispersible in polar and nonpolar media was carried out in micellar assemblies from amphiphilic invertible polyurethanes (AIPUs) based on poly(ethylene glycol) as a hydrophilic constituent and polytetrahydrofuran as a hydrophobic constituent. At a low concentration, AIPUs form micelles with a hydrophilic interior and a hydrophobic exterior. As concentration increases, individual micelles self-assemble into micellar assemblies with hydrophilic interior and hydrophobic exterior domains. It was shown that these domains can be applied simultaneously as nanoreactors and colloidal stabilizers to synthesize size-controlled batches of AgNPs in a nonpolar solvent. Size control and narrow particle size distribution can be facilitated by changes in the AIPU composition, macromolecular configuration, and polymer solution concentration. Depending on the length of polymer hydrophilic and hydrophobic fragments, and fragment distribution along the macromolecule, the size of fabricated AgNPs has been changed from 6 to 14 nm. Owing to the invertible properties of polyurethanes, the synthesized silver nanoparticles can be successfully dispersed in either polar or nonpolar media, where they form stable colloidal solutions.

Application of acoustic full waveform inversion to retrieve high‐resolution temperature and salinity profiles from synthetic seismic data

Recent works show that multichannel seismic (MCS) systems are able to provide detailed information on the oceans' fine structure. The aim of this paper is to analyze whether 1‐D full waveform inversion algorithms are suitable to recover the extremely weak acoustic impedance contrasts associated to the oceans' fine structure, as well as their potential to image meso‐scale objects such as meddies. We limited our analysis to synthetic, noise‐free data, in order to identify some methodological issues related to this approach under idealistic conditions (e.g., 1‐D wave propagation, noise‐free data, known source wavelet). We first discuss the influence of the starting model in the context of the multi‐scale strategy that we have implemented. Then we show that it is possible to retrieve not only sound speed but also salinity and temperature contrasts within reasonable bounds from the seismic data using Neural Network relationships trained with regional oceanographic data sets. Potentially, the vertical resolution of the obtained models, which depends on the maximum frequency inverted, is of the order of 5–10 m, whereas the root mean square error of the inverted properties is shown to be ∼0.5 m/s for sound speed, 0.1°C for temperature, and 0.06 for salinity. To conclude this study, we have inverted synthetic data simulated along an oceanographic transect acquired during the EU‐funded Geophysical Oceanography (GO) project. The results demonstrate the applicability of the method for synthetic data, as well as its potential to define oceanographic features along 2‐D transects at full ocean depth with excellent lateral resolution.

Finite character, local stability property and local invertibility property

We illustrate some results relating the finite character property, the local stability property and the local invertibility property of a domain and give a partial answer to two open questions.

Feedback Capacity of a Class of Symmetric Finite-State Markov Channels

We consider the feedback capacity of a class of symmetric finite-state Markov channels. Here, symmetry (termed “quasi-symmetry”) is defined as a generalized version of the symmetry defined for discrete memoryless channels. The symmetry yields the existence of a hidden Markov noise process that depends on the channel's state process and facilitates the channel description as a function of input and noise, where the function satisfies a desirable invertibility property. We show that feedback does not increase capacity for such class of finite-state channels and that both their nonfeedback and feedback capacities are achieved by an independent and uniformly distributed (i.u.d.) input. As a result, the channel capacity is explicitly given as a difference of output and noise entropy rates, where the output is driven by the i.u.d. input.

On Normal Forms of Nonlinear Systems Affine in Control

The nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory. The extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated. In this paper, we propose constructive algorithms for decomposing a nonlinear system that is affine in control. These algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs. They lead to various normal form representations and reveal the structure at infinity, the zero dynamics and the invertibility properties, all of which represent nonlinear equivalences of relevant linear system structural properties.

Invertibility and Non-Invertibility in Thin Elastic Structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E h. Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h −α E h for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.

A Geometric Approach to Zero Cancellation in Linear Multivariable Systems with Direct Feedthrough Terms

AbstractA geometric method for the design of a minimal-order dynamic feedforward compensator cancelling the minimum-phase invariant zeros of a linear multivariable system, while preserving the properties of controllability, observability, and right invertibility is discussed. The method is illustrated for continuous-time, non-strictly proper systems. Nonetheless, it also applies to strictly-proper systems.

On the nature of the laplace–beltrami operator on lipschitz manifolds

We study the basic properties of the Laplace–Beltrami operator on Lipschitz surfaces, as well as abstract Lipschitz manifolds, including mapping and invertibility properties on scales of Sobolev spaces, being the infinitesimal generator of an analytic semigroup, the nature of the spectrum and the regularity of eigenfunctions. Much of this analysis is carried out in the more general case of second order, divergence-form, strongly elliptic differential operators with bounded, measurable, complex matrix-valued coefficients. Bibliography: 34 titles.

Left invertibility of discrete systems with finite inputs and quantised output

The aim of this article is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems which evolve in discrete time within a continuous state-space. Quantised outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. Our main results are obtained under some contractivity hypotheses. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of iterated function systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system. Our main result is a necessary and sufficient condition for left invertibility and uniform left invertibility for joint contractive systems. In addition, an algorithm is proposed to recover inputs from output strings. A few examples are presented to illustrate the application of the proposed method.

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics

Mapping and invertibility properties of some parametrix-based surface and volume potentials are studied in Bessel-potential and Besov spaces. These results are then applied to derive regularity and asymptotics of the solution to a system of boundary-domain integral equations associated with a mixed BVP for a variablecoefficient PDE, in a vicinity of the curve of change of the boundary condition type.