Continuous Inverse Research Articles

L/sub p/ stability and linearization

A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Recently, under a typically weak additional assumption on the map, it was shown that Hadamard's condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. Here, an application is given concerning the relation between the L/sub p/ stability (with 1 /spl les/ p < /spl infin/) of a nonlinear system and the stability of related linear systems. We also give a result that directs attention to a fundamental limitation concerning what can be proved about linearization and stability for a related familiar family of feedback systems.

Stability and linearization: discrete‐time systems

AbstractA theorem by Hadamard gives a two‐part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Recently, under a typically weak additional assumption on the map, it was shown that Hadamard's condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. Here an application is given concerning the relation between the stability of a discrete‐time non‐linear system and the stability of related linear systems. Copyright © 2002 John Wiley &amp; Sons, Ltd.

A Partial Converse to Hadamard’s Theorem on Homeomorphism

A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Here, under a typically weak additional assumption on the map, we show that Hadamard’s condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. An application is given concerning the relation between the stability of a nonlinear system and the stability of related linear systems.

Bouligand dimension and almost Lipschitz embeddings

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dim b (A - A) denote the Bouligand dimension of the set A - A of differences between elements of A. Given any compact set A C R N such that dim b (A-A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P| A has Lipschitz continuous inverse except for a logarithmic correction term.

Dual-phase continuity and phase inversion in polycarbonate/polystyrene blends during compounding

Dual-phase continuity is a special case of morphologies in binary polymer blends where both phases maintain their whole continuity. Since this morphology can affect many physical and mechanical properties including impact strengh, elongational properties, and permeability characteristics [1–3], it is of great importance to know how these complicated morphologies are formed in situ and can be controlled during processing. However, the complexity of the microstructure brings about difficulties for its characterization and description. Until now, a few studies have been conducted in this area [4–6], but co-phase continuity remains one of the poorly understood phenomena in polymer blends during compounding. The nature of co-continuity and phase inversion after melt blending or after re-melt blending in twin-screw extruders has not been fully described. The effect of polymer/polymer blending in injection molding machines on the morphology is also unknown. In this study, the morphology of polycarbonate (PC) and polystyrene (PS) blends during compounding both in a twin-screw extruder and in an injection molding machine and the validity of the theoretical model in predication of the point of phase inversion were investigated in the composition range. A semi-empirical expression for predicting the point of phase inversion based on torque ratio has been proposed by Avgeropoulos et al. [7] expressed as:

Open Mappings of Probability Measures and the Skorokhod Representation Theorem

We prove that for the wide class of spaces X and Y (including completely regular Souslin spaces), every open surjective mapping f: $X\to Y$ induces the open mapping $\hat f$: $\mu\mapsto\mu\circ f^{-1}$ between the spaces of probability measures ${\cal P} (X)$ and ${\cal P} (Y)$. We discuss the existence of continuous inverse mappings for $\hat f$ and connections with the Skorokhod representation theorem and its generalizations.

Morphologies and Energies of Néel Inversion Wall Defects in a Liquid Crystal Polyether

Two-dimensional Néel inversion walls formed in a nematic monodomain during magnetic realignment of a highly elastically anisotropic thermotropic liquid crystal polyether (ε ≈ +0.5; k11 ≈ 3k33) are examined via polarized light microscopy (PLM). The detailed director patterns of these walls are then imaged at high resolution by atomic force microscopy (AFM) through a lamellar decoration technique, employed after the walls have been fixed by quenching while under a field. Walls form as closed loops composed of a continuous inversion wall with antiparallel director alignment of the interior region with respect to that of the exterior. The energy of Néel walls is theoretically evaluated as a function of elastic anisotropy, and it is shown that, in the extreme of ε = +1.0, indicating easy bend distortion, a Néel bend wall is 58% lower in energy than a Néel splay wall. Correspondingly, for easy splay distortion, i.e., ε = −1.0, the situation reverses. In our experimental system, the energy of a splay wall is 17% higher than that of a bend wall. The variation of the characteristic width of the walls in this polyether as measured by AFM yields an effective elastic constant of 1.6 × 10-6 dyn at 160 °C. Inversion wall dynamics can be followed through time-under-field experiments. Loops coalesce, shrink, and smooth their curvature, occasionally splitting into partial loops terminated by two opposite-strength 1/2 disclinations. The observed more rapid shrinkage of splay-distortion-rich wall segments as compared to that of bend-rich wall segments corresponds to the wall energetics expected on the basis of the elastic anisotropy of the polymer.

On quasimonotone increasing homeomorphisms

Let \( f\in C(\Bbb R^n,\Bbb R^n) \) be quasimonotone increasing such that \( \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) \) for a linear and strictly positive functional \( \Psi \) and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of \( x'=f(x) \).

An explicit right inverse of the divergence operator which is continuous in weighted norms

The existence of a continuous right inverse of the divergence operator in W 1,p 0 (Ω) , 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity holds also for some weighted norms. Our results are valid for Ω ⊂ R a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence and the classical theory of singular integrals of Calderon and Zygmund together with general results on weighted estimates proven by Stein. The weights considered here are of interest in the analysis of finite element methods. In particular, our result allows to extend to the three dimensional case the general results on uniform convergence of finite element approximations of the Stokes equations.

Relationship between plasma D(-)-lactate and intestinal damage after severe injuries in rats.

To explore the kinetic changes in plasma D(-)-lactate and lipopolysaccharide (LPS) levels, and investigate whether D(-)-lactate could be used as a marker of intestinal injury in rats following gut ischemia/reperfusion, burn, and acute necrotizing pancreatitis (ANP). Three models were developed in rats: (1)gut ischemia/reperfusion obtained by one hour of superior mesenteric artery occlusion followed by reperfusion; (2)severe burn injury created by 30% of total body surface area (TBSA) full-thickness scald burn; and (3)ANP induced by continuous inverse infusion of sodium taurocholate and trypsin into main pancreatic duct. Plasma levels of D(-)-lactate in systemic circulation and LPS in portal circulation were measured by enzymatic-spectrophotometric method and limulus amebocyte lysate (LAL) test kit, respectively. Tissue samples of intestine were taken for histological analysis. One hour gut ischemia followed by reperfusion injuries resulted in a significant elevation in plasma D(-)-lactate and LPS levels, and there was a significant correlation between the plasma D(-)-lactate and LPS (r = 0.719, P<0.05). The plasma concentrations of D(-)-lactate and LPS increased significantly at 6h postburn, and there was also a remarkable correlation between them (r=0.877 P<0.01). D(-)-lactate and LPS levels elevated significantly at 2h after ANP, with a similar significant correlation between the two levels (r = 0.798, P < 0.01). The desquamation of intestine villi and infiltration of inflammatory cells in the lamina propria were observed in all groups. The changes of plasma D(-)-lactate levels in systemic blood paralleled with LPS levels in the portal vein blood. The measurement of plasma D(-)-lactate level may be a useful marker to assess the intestinal injury and to monitor an increase of intestinal permeability and endotoxemia following severe injuries in early stage.

Extension of entire functions of completely regular growth and right inverse to the operator of representation of analytic functions by quasipolynomial series

Let be an entire function of one complex variable that has exponential type, completely regular growth, and whose conjugate diagram is equal to the sum of the closure of a bounded convex domain and a convex compact subset of . Criteria ensuring that the operator of the representation of analytic functions in by quasipolynomial series with zeros of the function as exponents has a continuous linear right inverse are established. These criteria are stated in terms of conformal maps of the unit disc onto the domain and of the exterior of the closed unit disc onto the exterior of , and of extensions of the original function to an entire function of two complex variables whose absolute value satisfies certain (upper) estimates. An analogue of the Leont'ev interpolation function defined by this extension is used to obtain formulae for the continuous linear right inverse to the representation operator .

Determination of the aerosol size distribution by analytic inversion of the extinction spectrum in the complex anomalous diffraction approximation

A new derivation is presented for the analytical inversion of aerosol spectral extinction data to size distributions. It is based on the complex analytic extension of the anomalous diffraction approximation (ADA). We derive inverse formulas that are applicable to homogeneous nonabsorbing and absorbing spherical particles. Our method simplifies, generalizes, and unifies a number of results obtained previously in the literature. In particular, we clarify the connection between the ADA transform and the Fourier and Laplace transforms. Also, the effect of the particle refractive-index dispersion on the inversion is examined. It is shown that, when Lorentz's model is used for this dispersion, the continuous ADA inverse transform is mathematically well posed, whereas with a constant refractive index it is ill posed. Further, a condition is given, in terms of Lorentz parameters, for which the continuous inverse operator does not amplify the error.

Characterization of the homogeneous polynomials P for which (P + Q)(D) admits a continuous linear right inverse for all lower order perturbations Q

Characterization of the homogeneous polynomials P for which (P + Q)(D) admits a continuous linear right inverse for all lower order perturbations Q

Local inverses of Borel homomorphisms and analytic P‐ideals

We present a theorem on the existence of local continuous homomorphic inverses of surjective Borel homomorphisms with countable kernels from Borel groups onto Polish groups. We also associate in a canonical way subgroups of ℝ with certain analytic P‐ideals of subsets of ℕ. These groups, with appropriate topologies, provide examples of Polish, nonlocally compact, totally disconnected groups for which global continuous homomorphic inverses exist in the situation described above. The method of producing these groups generalizes constructions of Stevens and Hjorth and, just as those constructions, yields examples of Polish groups which are totally disconnected and yet are generated by each neighborhood of the identity.

The association between hospital volume and survival after acute myocardial infarction in elderly patients.

Patients with chest pain thought to be due to acute coronary ischemia are typically taken by ambulance to the nearest hospital. The potential benefit of field triage directly to a hospital that treats a large number of patients with myocardial infarction is unknown. We conducted a retrospective cohort study of the relation between the number of Medicare patients with myocardial infarction that each hospital in the study treated (hospital volume) and long-term survival among 98,898 Medicare patients 65 years of age or older. We used proportional-hazards methods to adjust for clinical, demographic, and health-system-related variables, including the availability of invasive procedures, the specialty of the attending physician, and the area of residence of the patient (rural, urban, or metropolitan). The patients in the quartile admitted to hospitals with the lowest volume were 17 percent more likely to die within 30 days after admission than patients in the quartile admitted to hospitals with the highest volume (hazard ratio, 1.17; 95 percent confidence interval, 1.09 to 1.26; P<0.001), which resulted in 2.3 more deaths per 100 patients. The crude mortality rate at one year was 29.8 percent among the patients admitted to the lowest-volume hospitals, as compared with 27.0 percent among those admitted to the highest-volume hospitals. There was a continuous inverse dose-response relation between hospital volume and the risk of death. In an analysis of subgroups defined according to age, history of cardiac disease, Killip class of infarction, presence or absence of contraindications to thrombolytic therapy, and time from the onset of symptoms, survival at high-volume hospitals was consistently better than at low-volume hospitals. The availability of technology for angioplasty and bypass surgery was not independently associated with overall mortality. Patients with acute myocardial infarction who are admitted directly to hospitals that have more experience treating myocardial infarction, as reflected by their case volume, are more likely to survive than are patients admitted to low-volume hospitals.

Quelques nouvelles identités de fluctuation pour les processus de Lévy

Let τ and H be the ladder time and ladder height processes of a Lévy process X. We give an identity in law between (τ,H) and (X,H *), H * being the right continuous inverse of the process H. The later allows us to get a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (X t- inf s≤ X s >- 0). In the stable case, some explicit calculations are provided.

Simultaneous filtration and immobilization of cells from a flowing suspension using a bioreactor containing polyethylenimine coated cotton threads: Application in the continuous inversion of concentrated sucrose syrups

Polyethylenimine(PEI)-coated cotton threads were shown to have potential for reducing microbial load from a flowing suspension. Turbid cell suspensions perfused through the PEI column appeared as totally clear in the effluent. The adhesion efficiency of the matrix was found to depend on the concentration of PEI used to treat the threads. Threads coated with 2.5% PEI were found to show optimal retention of cells. A considerable amount of binding was seen over a broad range of ionic concentration (0–0.3 M) and pH (3.6–10.3). Under similar conditions control threads did not show any filtration capacity. Saccharomyces cerevisiae, Saccharomyces fragilis, Escherichia coli and an Acetobacter species could be effectively filtered using PEI-coated threads. This technique can find potential for the simultaneous filtration and immobilization of cells in a bioreactor to be used in continuous bioprocessing as exemplified for the inversion of sucrose syrups using baker's yeast. The bioreactor could continuously hydrolyse 60% (w/v) sucrose syrups with a productivity of 2.25 kg/day for over a month without loss in efficiency.

Phragmén-Lindelöf principles on algebraic varieties

Estimates of Phragmén–Lindelöf (PL) type for plurisubharmonic functions on algebraic varieties in C n \mathbb {C}^n have been of interest for a number of years because of their equivalence with certain properties of constant coefficient partial differential operators; e.g. surjectivity, continuation properties of solutions and existence of continuous linear right inverses. Besides intrinsic interest, their importance lies in the fact that, in many cases, verification of the relevant PL-condition is the only method to check whether a given operator has the property in question. In the present paper the property P L ( R n , ω ) {PL}({\mathbb R}^n,\omega ) which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical Phragmén–Lindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between P L ( R n , ω ) {PL}({\mathbb R}^n,\omega ) and the PL-condition used by Hörmander to characterize the surjectivity of differential operators on real-analytic functions. We prove the property P L ( R n , ω ) {PL}({\mathbb R}^n,\omega ) for an algebraic variety V V implies that V h V_h , the tangent cone of V V at infinity, also has this property. The converse implication fails in general. However, if V h V_h is a manifold outside the origin, then V V satisfies P L ( R n , ω ) {PL}({\mathbb R}^n,\omega ) if and only if the real points in V h V_h have maximal dimension and if the distance of z ∈ V z\in V to V h V_h is bounded by C ω ( | z | ) C\omega (|z|) as z z tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy P L ( R n , ω ) {PL}({\mathbb R}^n,\omega ) is known, nor any of the other PL-conditions alluded to above. Besides these main results the paper contains several auxiliary necessary conditions and sufficient conditions which make it possible to treat interesting examples completely. Since it was submitted they have been applied by several authors to achieve further progress on questions left open here.

Nichtlineare stetige Beobachter

AbstractMost state estimation methods for smooth nonlinear systems are also smooth and require that the Taylor linearization of the system is observable. The presented design of continuous observers can be applied if the nonlinear system is observable but not its Taylor linearization. In this case, the observability map of the system is a semi‐diffeomorphism which means that the map has only a continuous (and no smooth) inverse. A continuous observer is designed in the observability normal form coordinates and comprises a dynamic and an algebraic part. At least the algebraic part defined by the inverse observability map is continuous.The continuous observer approach can also be used to solve the critical observation problem if the Taylor linearization has only non‐detectable eigenvalues with a zero real part. Moreover, this approach allows an extension of the nonlinear normal form observer design because the observability rank condition is no more required. As distinct from the first presentation of continuous observers an [9], the design of nonlinear continuous observers is treated from a more applied point of view and is illustrated by examples for which smooth observers do not exist.

The discrete Radon transform and its approximate inversion via linear programming

Let S be a finite subset of a lattice and let v s ( L), the number of points of S ∩ L for each line L, denote the discrete Radon transform of S. The problem is to reconstruct S from a knowledge (possibly noisy) of the restriction of v s to a subset L of the set of all lines in any of a few given directions through the lattice. Reconstructing a density from its line integrals is a well-understood problem, but discreteness causes many difficulties and precludes use of continuous Radon inversion algorithms. Indeed it has been shown that when the directions are main directions of the lattice, the case for most applications, the problem is finite but is NP-hard, so that any reconstruction algorithm will surely have to consist of exponentially many steps in the size of S. We address this problem by looking instead for a fuzzy set S with the given line sums, i.e. a function f( z) with 0 ⩽ f( z) ⩽ 1 for all points z in the lattice, for which Vf( L)  v s ( L). The set of all such f forms a convex set and those f = χs with each f( z) ϵ {0, 1} are extreme points. Finding a fuzzy set f with the given line sums is a linear programming problem and so there are efficient algorithms for finding f or proving that no such f (and hence no S) exists with the given line sums. If S is an additive set with respect to L , i.e. if we can write S = { z ∑ LϵLϵz g( L) > 0} for some functional g on L , we show that there is only one fuzzy set f with the given line sums. We prove here that if S is not additive then there are many fuzzy sets with the given line sums, although there still may be only one actual set. Linear programming methods that are based on interior point methods always produce solutions that lie in the center of the convex set of all solutions. As a result, if S is a set with given line sums and linear programming produces a solution that consists only of {0, 1} then this solution must be the original subset S, and S must be a set of uniqueness. Thus interior point LP's give a polynomial and practical way to obtain the assertion of uniqueness when strong uniqueness obtains. This problem arises in a practical situation, although it was earlier studied in the case of coordinate directions for its intrinsic interest. In the practical situation, S represents a piece of a real crystal, and the line sums in any fixed direction can be measured (possibly with uncertainties) using a transmission electron microscope. In order to study the behavior of the linear programming reconstruction algorithm and to explore the question of whether a “typical” set S with a “typical” L will be a case of uniqueness or near-uniqueness, we conducted simulation experiments. These indicate that the method is practical and reasonably efficient, at least on the examples we considered.