Real Radius Research Articles

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae.

The Historical Re-Brightening and Distance Recheck of SN 1006

We presume the re-brightening of SN 1006 in AD 1016 as recorded in the ancient Chinese literature to be true and the re-brightening was caused by the encounter either of photons or the shock wave from the SN outburst with the circumstellar thin envelope materials. Based on these considerations, and combining the observational results on the optical proper motion of the N-W limb and the radio observations of the other parts of the supernova remnant G327.6+14.5, we re-determine the distance to SN 1006. For the photon-encounter model, the average radius of the envelope material would be 10 ly; and for the shock wave-encounter model, the radius would be about 1 ly. We then set up four equations to solve for the distance of the SN, the initial shock speed, the expansion index for two different parts of the supernova remnant, and the real original radius of the thin envelope nebula. It is indicated that only the case of photon-encounter will lead to a reasonable result. We derived a distance of 5074 ly (1.56 kpc), an original shock expansion velocity of 0.071c, an expansion index of 0.72 for the N-W limb of the SNR, and 0.76 for the other parts . We deem that the SNR evolution is still in the stage of reverse shock.

A Precision Tension Control System Based on PIC

As an important factor in composite fiber winding, fiber winding tension influences the product quality directly, and therefore tension control is a key technique in fiber winding techniques. This paper introduces a closed loop tension control system with the PIC single chip computer as the kernel, a function module with TLC5615 D/A converter as the output, AC servomotors as the actuator, and a radius-following arm to measure the real time radius of the scroll. The mechanism of the tension control system is analyzed and the mathematical model is set up. Then the paper expatiates the design of main control circuit, the voltage transformation circuit and the analogy filter circuit. In addition, the paper presents digital filter and control algorithm of the PIC software. The system is proved to have high control precision and rapid time response through performance experiments of the tension control system in different working states, which well meets the anticipated design requirements.

Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying $$ \Delta ^{{II}} \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} = A\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} $$

In this paper the surfaces of revolution without parabolic points, in the 3-dimensional Lorentz-Minkowski space are classified under the condition $$ \Delta ^{{II}} \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} = A\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{r} $$ where Δ II is the Laplace operator with respect to the second fundamental form and A is a real 3 × 3 matrix. More precisely we prove that such surfaces are either minimal or Lorentz hyperbolic cylinders or pseudospheres of real or imaginary radius.

Strong robustness measures for sets of linear SISO systems

In this paper the issue of strong robustness of sets of systems is introduced and motivated in a scenario of adaptive control. A set of systems is said to be time-invariant strongly robust if the optimal control law designed on the basis of any element in this set stabilizes any other fixed member of the set. A stronger notion amounts to requiring that time-varying stabilizing control laws designed on the basis of any sequence of elements in the set asymptotically or quadratically stabilizes any other fixed member of the set. In this case, the set of systems is said to be time-varying strongly robust or time-varying strongly quadratically robust. The framework is exhibited for a class of linear and time-invariant SISO systems in discrete-time with fixed order n for a specified class of control objectives. It is established that if a given set of systems satisfies some robustness measures involving the classical structured real or complex radius, then the set is strongly robust. Then, for a more specific class of control objectives, attention is paid to the case of polyhedral sets of systems. A characterization of time-varying strongly quadratically robust sets of systems is given in the form of a feasibility test on a finite set of linear matrix inequalities (LMI's).

Linear operators preserving the sign-real spectral radius

Define the sign-real spectral radius of a real n× n matrix A as ρ 0 s(A)= max S∈ S ρ 0(SA) , where ρ 0( A)=max{∣ λ∣; λ a real eigenvalue of A} is the real spectral radius of A and S denotes the set of signature matrices, i.e. S={S;∣S∣= I} , the absolute value of matrices being meant entrywise. In this paper we show that linear invertible operators F on the space of n× n real matrices M n( R) preserving the sign-real spectral radius are exactly the operators of the form F( A)= P T D −1 SA (T) DP with P a permutation matrix, D a diagonal matrix and S a signature matrix.

Gap dependence of the tip-sample capacitance

The tip-sample capacitance has been studied in the nontunneling regime and the capacitance-distance characteristics and its dependence on the tip geometry have been determined for the gap distance 1<s<600 nm. Measurements were carried out in ultrahigh vacuum on a capacitor formed between a metal tip (W or Pt–Ir) and a clean Au(111) surface. Tips of different tip radius R=30∼4000 nm were used to investigate the influence of tip geometry on the capacitance. When the gap distance is reduced, the capacitance increases while its gap sensitivity |∂C/∂s| decreases with the gap distance. The capacitance therefore shows no 1/s divergence. The magnitude of the capacitance change is found to depend on the tip geometry: blunt tips (R>1000 nm) show larger capacitance increase than that for sharp tips (R⩽100 nm). The effective tip radius Reff estimated from the C−s characteristics agrees with the real tip radius in a limited distance range which varies with the tip geometry. At small distances (s<30 nm), Reff≃R for sharp tips but Reff<R for blunt tips. On the other hand at large distances (s>200 nm), the relation is reversed, Reff>R for sharp tips while Reff∼R for blunt tips. These results on Reff can be explained by the field concentration to the tip apex and the change of capacitance-contributing tip area with the gap distance. Capacitance calculations indicate that the capacitance of the “truncated cone + half sphere” tip well reproduces the observed C−s characteristics and its dependence on the tip geometry.

A Formula for the Stability Radius of Time-Varying Systems

This paper considers the stability radius of time-varying systems with respect to linear dynamical perturbations. A formula for the stability radius in terms of the norm of a certain input–output operator is developed. Further it is shown that the real and complex stability radius coincide.

COMPLEX AND REAL PERFORMANCE RADII AND THEIR COMPUTATION

This paper considers the problem of robust performance of a linear time-invariant system in /spl Hscr//sub /spl infin// norm. The concepts of complex and real performance radii are introduced to describe the smallest size of dynamic or parametric perturbations to a feedback system that either destabilize the system or destroy a performance bound in certain closed loop transfer matrix of the system. An algorithm to compute the complex performance radius is given. For the real performance radius, a lower bound, which often turns out to be exact, is obtained.

On the computation of the real Hurwitz-stability radius

Recently Qiu et al. (1995) obtained a computationally attractive formula for the evaluation of the real stability radius. This formula involves a global maximization over frequency. Here, for the Hurwitz stability case, we show that the frequency range can be limited to a certain finite interval. Numerical experimentation suggests that this interval is often reasonably small.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of the relationships between Junge size distribution and Ångström α turbidity parameters from spectral measurements of atmospheric aerosol extinction

The Junge aerosol size parameter ν has been determined from visible spectral solar extinction data. Two different methods have been used: the first is based on linearizing the Ångström formula along with the well-known relationship ν= α + 2. The second, more complex, method uses the Mie scattering theory in an aerosol atmospheric model with a single attenuation layer. Different turbidity conditions have been considered. The obtained ν values from the two methods show a high level of disagreement. Therefore, a full analysis between Junge and Ångström α parameters with real and complex refractive indices and different particle radius intervals has been carried out. In this context, the applicability limits of the Junge relationship are shown in order for application under real atmospheric turbidity conditions.

High-frequency nonlinear acoustic beams and wave packets

Gaussian beams and wave packets are formally equivalent to high-frequency solutions of the linear wave equation with complex-valued phase functions. The theory of weakly nonlinear high-frequency waves is extended in this paper to allow for complex phase solutions. The procedure is similar to nonlinear geometrical optics and the nonlinearity causes the phase to vary, leading to the development of weak shocks. However, unlike standard geometrical optics that describes the evolution of wave fronts, complex phase solutions correspond to Gaussian decay away from a central ray, and the associated curvatures are complex valued. Geometrical singularities due to caustics and foci do not occur, and the only singularities in the theory are purely nonlinear. The theory is developed specifically for nonlinear acoustic waves in a homogeneous inviscid fluid. The time taken for nonlinear singularities to develop is compared for a real spherical wave front and a Gaussian beam with Gaussian radius equal to the real radius of curvature of the wave front. The time to blowup for the Gaussian beam is intermediate between the shortest possible, which is for the converging wave front, and the longest possible, for the diverging front.

Permeability of Unidirectional Reinforcements for RTM

The permeability of an idealized unidirectional reinforcement consisting of regularly ordered, parallel fibres is derived starting from first principles (Navier-Stokes equations) both for flow along and for flow perpendicular to the fibres. First, an approx imate analytical solution for transverse flow is derived which differs from the Kozeny- Carman equation for the permeability of a porous medium [9] in that the transverse flow stops when the maximum fibre volume fraction is reached. The solution for flow along the fibres has the same form as the Kozeny-Carman equation. A comparison shows excellent agreement between a numerical solution of the full flow equations and the approximate one at medium to high fibre volume fractions ( V f &gt; 0.35). The theoretical predictions of permeability were tested in a specially designed mould. The results from the experiments with an unsaturated polyester resin (Jotun PO-2454) and the unidirectional reinforcement did in all cases show excellent agreement with results predicted by Darcy's law (the square of the flow front position increases linearly with time if the injection pressure is kept con stant). The theoretical model could be fitted to the experimental data both for flow along the fibres and for cross flow based on data for flow along the fibres only. The fitting is ob tained by adjusting one parameter in the model, the effective fibre radius, to a value about four times larger than the real fibre radius (15 μm). Scanning electron microscopy shows that the fibres are arranged in bundles looking like cylinders with ellipsoidal cross section which may be the explanation for the effective fibre radius in the fitted model equation be ing larger than the real fibre radius.

Isotopic effects and surface absorption in 35,37Cl+24Mg interactions

The few-nucleon transfer is found to play an important role in the isotopic effects observed in absorption. This conclusion is obtained by measuring elastic scattering and quasielastic reactions and by analysing elastic data with both phenomenological and microscopic models. The sensitivity domain is found to be different for imaginary and real potentials. The implication for the validity of the dispersion relation for phenomenological potentials at the real sensitivity radius, when transfers are important, is discussed.

Digital circles with non-lattice point centers

Digitization of a real circle is defined as the set of closest pixels (integer lattice points) to the circle. These pixels form a digital circle. First, this paper defines the term “closest pixels” and provides an algorithm to digitize an arbitrary circle, i.e., one whose radius is not necessarily an integer, and whose center is not necessarily a lattice point. This digitizing algorithm is simplified to the case of integer radius and integer-coordinate center so that it can be compared to two well-known algorithms: the Bresenham algorithm and an algorithm using the Digital Differential Analyzer (DDA) technique. As a result, the new algorithm consumes only 77% and 40% of the time required for Bresenham and DDA algorithms, respectively. Second, we investigate the problem of finding, for a given digital circle and center, the set of possible real radii that generate that digital circle. For some integerr, we look at all digital circles centered at the origin with real radius in the interval [r−0.5,r+0.5) and determine the paritions of this interval into sub-intervals of real radii generating the same digital circles. Third, we study fixing the digital circle and radius, and finding the set of possible centers. For some pixel (xc,yc), we find the partitions of the unit square [x c −0.5,x c +0.5)×[y c −0.5,y c +0.5) such that for each partition each center in that partition generates the same digital circle. The partition boundaries can be constructed by compass and ruler. Finally, the paper uses these results to determine whether or not a given finite set of pixels is a digital circle segment and if so, shows how to construct its region of possible centers by ruler and compass. We also present an algorithm to detect whether a closed digital are is a digital circle.

A note on some differences between real and complex stability radii

Real and complex stability radii measure the robustness of stable linear systems under real and complex parameter perturbations, respectively. In this note we point out some basic differences between them. In particular, we investigate their different behaviour with respect to system interconnections and changes in system data.

Stability radii of linear systems

In this note the concepts of real and complex stability radii are introduced and some of their properties discussed.

On the conformity of soft lenses to the shape of the cornea.

The assumption that a properly fitted soft contact lens conforms to the shape of the cornea in the central region was tested by measuring the apparent radius of curvature of the back surface of various soft contact lenses on the eye, with a high illumination keratometer. Real radius of curvature was calculated by means of an appropriate formula. It was found that soft lenses conform closely to the corneal shape. The mean difference between the radius of the back surface of the lens and corneal radius of curvature was +0.015 mm in the horizontal meridian, and +0.023 mm in the vertical meridian. The lenses were slightly flatter than the cornea in each meridian. The difference in conformity between the horizontal and vertical meridians was not significant. Slight differences were found in the degree of conformity of the three soft contact lens designs studied. The thicker, lower water content lenses showed slightly less conformity.

A global optical-model analysis of neutron elastic scattering data

Abstract Differential cross sections for neutron elastic scattering in the energy range 7–26 MeV from 40Ca, 90Zr, 92Mo, 116, 124Sn and 208Pb are analyzed in terms of a global optical-model potential. A smooth variation of the real radius parameters with mass number is investigated. Otherwise, optical-model geometrical parameters are kept fixed at values obtained by averaging individual best fits. The energy and isospin dependences of potential strengths and volume integrals per nucleon are presented.

Conformation of poly(isoprene‐gstyrene) in dilute solution

AbstractA detailed description of the synthesis of a model poly(isoprene‐g‐styrene) is given. From light scattering measurements the real radius of gyration of the polyisoprene is backbone is found to be smaller than that of the polystyrene branches. The core‐and‐shell conformation of graft copolymers in dilute solution in good solvents, which was proposed for poly(styrene‐g‐isoprene), is supported by these results.