Classical Plane Research Articles

Absolute Points of Continuous and Smooth Polarities

This paper deals with sets of absolute points of continuous or smooth polarities in compact, connected or smooth projective planes, called topological polar unitals or smooth polar unitals, respectively. We will show that topological polar unitals are Z2-homology spheres. In the four-dimensional case, a topological polar unital U is either a topological oval, or any line which intersects U in more than one point intersects in a set homeomorphic to S1. Smooth polar unitals turn out to be smoothly embedded submanifolds of the point space. Moreover, secants intersect such unitals transversally. For these unitals, we will obtain full information on the existence of secants, tangents and exterior lines through given points according to their position with respect to the unital. The main result of this paper states that the possible dimensions of smooth polar unitals coincide with those of sets of absolute points of continuous polarities in the classical projective planes P2F, Fϵ{R,C,H,O}. Finally, we will prove that smooth polar unitals in four-dimensional smooth projective planes are topological ovals or are homeomorphic to S3.

Acoustic Doppler current profiler measurements of the velocity field of an ebb tidal jet

A broadband acoustic Doppler current profiler was used to measure the velocity field of the ebb tidal jet formed at the entrance to the Otago Harbour, South Island, New Zealand. Four separate surveys were conducted where various sets of transects were run continuously over a 13 hour period, producing time series of velocity profiles with 1 hour sampling over a grid with an across‐jet resolution of 20 m and an along‐jet resolution of 150 m. The velocity field of the ebb tidal jet is phase locked to the tidal response of the harbor, making it possible to combine the data from separate tidal cycle surveys to produce synoptic maps of the depth‐averaged velocity field. The spatial structure of the ebb tidal jet at the time of peak flow exhibits many features of a classical plane jet. There is a strong asymmetry between the ebb and flood flow structures; the ebb flow extends beyond 2 km from the entrance, while the flood flow is limited to within 600 m of the coast. The flow along the jet axis becomes pulse‐like in time with distance from the entrance. The pulse that develops propagates way from the entrance at a speed slower than the advective speed of the ebb jet. The formation of a propagating eddy observed along the eastern side of the jet suggests that this momentum pulse may be associated with the pair of separation eddies generated by the accelerating ebb tidal flow.

Modelling of elastic continua using a grillage of structural elements based on discrete element concepts

AbstractA framework is described for modelling an elastic continuum using a grillage of beam‐like structural elements derived from discrete element concepts. The beam element properties are derived in detail and implemented in a structural analysis code for validation against classical two‐dimensional plane elasticity solutions. The framework offers the possibility of modelling the onset and propagation of fracture in materials that are initially continuous, without the need for specialized elements or remeshing in the context of traditional finite elements. Copyright © 2001 John Wiley & Sons, Ltd.

Influence of a swirl motion on the interaction between microalgal cells and environmental medium during ultrafiltration of a culture of Tetraselmis suecica

The role of the environmental medium of a marine microalga culture (Tetraselmis suecica) in membrane fouling phenomenon is quantified for two ultrafiltration units: one generating a classical tangential plane flow and another involving a swirling decaying flow induced by a tangential inlet. Compared to the plane unit, the swirling configuration increases the performances of the ultrafiltration process (augmentation of 20% in terms of limiting flux). Interactions between the culture medium and cells depend on the module design. Furthermore, experiments emphasize the significant role of the particles whatever the module design: more than 70% of the total hydraulic resistance could be due to the microalgal cells, while about 30% of membrane fouling is relevant to soluble materials.

Two-beam interferometers: a classification which takes into account multiple localizations

Summary Two beam interferometers are traditionally classified by the method used to separate the beams. This classification is suitable for considering localizations when the source is incoherent and continuous since an amplitude division interferometer yields the classical localization plane while a wavefront division one yields no fringes. However when the source is incoherent and periodic and the symmetry is plane there can be multiple localization planes and these cannot be easily analysed using this classification. In the present paper these planes are taken into account classifying two beam interferometers as those where the ‘effective interfering sources’ are the images of the source of light and those where they are not. The latter either yield no fringes or non-localized ones while the former yield several planes with straight, sinusoidal, localized fringes provided certain requirements are fulfilled. One of these is the equivalent sine condition which is here derived for any observation plane. Thus interferometers of the first class are further subdivided in two: those where the equivalent sine condition is verified on every observation plane and those where it is not. Experimental results illustrating the validity of the theoretical predictions are shown.

Design of high finesse, wideband Fabry-Pe´rot filter based on chirped fiber Bragg grating by numerical method

The spectral characteristics of a Fabry-Perot spectrometer (filter) formed by a pair of identical linearly chirped fiber Bragg gratings in optical fiber are studied numerically using the characteristic matrix method. The results indicate that based on available techniques and materials, one can fabricate such filters with a finesse as high as 104 and contrast as high as 109 by inscribing a pair of identical linearly chirped fiber gratings. Achieving such superfinesse and contrast in a conventional Fabry-Perot (FP) is very difficult because of fabrication complexities in achieving superflatness at ultrahigh reflectivities simultaneously and keeping them aligned. Our numerical simulation also indicates that the spectral characteristics of the fiber FP (FFP) filter can be approximated by that of a classical plane mirror FP (PFP) with a mirror separation of L + ?L, where L is the length of any one of the two gratings and ?L is the separation between the two gratings. This analogous characteristic enables one to estimate time domain and other behavior of an FFP from already established PFP analysis. Thus, miniature FFPs can be used not only to achieve ultralow, crosstalk in wavelength division multiplexing (WDM), but they can also be integrated into miniature, hybrid, spectral sensors (such as Brillouin and Raman sensors) where ultrahigh contrast with superresolution is required.

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane \(\) the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space \(\) thus defining a locally compact topological affine translation plane \(\). We introduce the moduli space \(\) of isomorphism classes of compact spreads, \(\). We show that for \(\) the topology of \(\) is not \(\) by constructing a sequence of non-classical spreads in \(\) that converges to the classical spread in \(\), where \(\). Moreover, we prove that the isomorphism type of \(\) varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes.

Implicit Characterizations of Smooth Incidence Geometries

We give a characterization of smooth stable and smooth projective planes in terms of submersion and transversality. Moreover, smooth affine translation planes are characterized by properties of their corresponding spreads, considered as subsets of the Grassmannian. The last section contains smoothness results about spherical Moebius planes. In particular, we establish smoothness properties of the classical Moebius plane.

A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory

The article describes and proves D. A. Grave's method that solves classical plane boundary-value problems for the Green's function of the Laplace equation in regions whose boundaries are smooth analytical curves defined by finite-order irreducible polynomials. The proposed method has certain advantages compared with the method that constructs the Green's function by conformally mapping the original region onto the unit disk. A class of regions are identified for which Grave's methods produces an explicit analytical solution in convergent-series form. This is a natural generalization of the conformal mapping method for simplest regions.

Fiber Element for Cyclic Bending and Shear of RC Structures. I: Theory

After a few years of successful application of the fiber beam element to the analysis of reinforced concrete (RC) frames, the introduction of the mechanisms of shear deformation and strength appears to be the next necessary step toward a realistic description of the ultimate behavior of shear sensitive structures. This paper presents a new finite-beam element for modeling the shear behavior and its interaction with the axial force and the bending moment in RC beams and columns. This new element, based on the fiber section discretization, shares many features with the traditional fiber beam element to which it reduces, as a limit case, when the shear forces are negligible. The element basic concept is to model the shear mechanism at each concrete fiber of the cross sections, assuming the strain field of the section as given by the superposition of the classical plane section hypothesis for the longitudinal strain field with an assigned distribution over the cross section for the shear strain field. Transve...

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers $\Bbb C $ , the quaternions $\Bbb H $ , and the Caley numbers $\Bbb O $ . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension $d \geq 17$ is isomorphic to the quaternion projective plane ${\cal P _2\Bbb H }$ . For topological compact projective planes this is true if $d \geq 19$ , and this bound is sharp.

Transfer matrix of curved duct bends and sound attenuation in curved expansion chambers

An explicit formula of the transfer matrix is derived for evaluating the acoustic performance of the chamber including curved duct bends. Based on the derived transfer matrix, the reactive sound attenuation of the chamber can be easily calculated by using an empirical formula for estimating the normalized maximum radial pressure of the fundamental mode in the curved duct bend. Predicted results are compared with experimental ones and they show good agreements. It is found that the classical plane wave theory for the straight duct of equivalent length to the curved duct bend is not applicable if the normalized maximum pressure is greater than 1.3. In addition, the sound attenuation in curved expansion chamber is investigated by studying the effect of geometrical parameters. The derived transfer matrix for the curved duct bend can be easily incorporated as a basic module into any existing computer programs dealing with four-pole parameters for predicting the acoustic performance of HVAC and other silencing systems.

On 4-dimensional elation Laguerre planes admitting simple Lie groups of automorphisms

This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit large automorphism groups. In particular, it is shown that such a plane is classical if its automorphism group is at least 11-dimensional. Furthermore, the elation Laguerre planes admitting simple Lie groups of automorphisms are investigated and various characterizations of the classical complex Laguerre plane and the semi-classical Laguerre planes are obtained.

Stabilizers of collineation groups of smooth stable planes

Let S be a smooth stable plane of dimension n (see Definition 1.2) and let Δ be a closed subgroup of the collineation group of S which fixes some point p. We derive some results on the group-theoretical structure of Δ, e.g. that Δ is a linear Lie group (Theorem 3.7). As a by-product this shows that no (affine or projective) Moulton plane can be turned into a smooth plane. If Δ fixes some flag, then any Levi subgroup Ψ of Δ is a compact group and Δ is contained in the flag stabilizer of the classical Moufang plane of dimension n (Corollary 3.1 and Theorem 3.7). Let Δ fix three concurrent lines through the point p. If S is one of the classical projective planes over the reals, the complex numbers, the quaternions, or the Cayley numbers, then the dimension of Δ is d class = 3, 6, 15, or 38, respectively. We show that for a smooth stable (projective) plane S of dimension 2l either S is an almost projective translation plane (classical projective plane) or that dim Δ ≤ d class − l holds (Theorems 4.1 and 4.2).

Analysis of electrojet-distorted magnetotelluric sounding curves

South America presents several unique geomagnetic features, one of which is the Equatorial Electrojet (EEJ), a current system which extends itself east to west in Northeastern Brazil for almost 3500 km. Considering the fact that the influence of this phenomenon at low magnetic latitudes can be detected at great distances from its central axis, it is important to simulate its effect in magnetotelluric (MT) exploration. To accomplish this and by the use of an integral equation scheme, we have modeled the MT response of confined three-dimensional (3D) bodies (dykes in an homogeneous half-space) and deep 3D structures (horst and graben located at Marajo Basin in Northern Brazil). As the inductive source, we replace the classical plane wave source by a common line of current, besides gaussian and undulatory electrojets. The results of the modelling indicate that the studied effect is prominent in both one-dimensional (1D) and 3D media for periods ( T) greater than 10 s. It decreases with distance, but it is detectable as far as 3000 km from the center of the EEJ. It is also observed that for T greater than 10 s, the computed soundings can be strongly distorted, mainly by EEJ effects in the host medium which, in turn, cause changes in the final branch of the 3D soundings. For structures in the Marajo Basin, our results reveal that the 3D galvanic effect prevails in the interval 0.1< T<10 s, while source influence occurs mainly for T>10 s. On the other hand, automatic inversion of these 3D data shows that, in equatorial regions having complex geology, some errors arise in conventional 1D interpretation of the MT soundings. This is due to the superposition of the host medium response, the galvanic effect of 3D structures, and the EEJ influence.

A common axiom set for classical and intuitionistic plane geometry

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

Noncommutative quantum conformal plane wave deformation

We construct a q-deformation of the plane wave as a formal power series in the noncommutative coordinates of q-Minkowski space-time and four-momenta. This q-plane wave has analogous properties to the classical one, in particular, it has the properties of q-Lorentz covariance, and it satisfies the q-d'Alembert equation (proposed earlier by one of us) on the q-Lorentz covariant momentum cone. On the other hand, this q-plane wave is not an exponent or q-exponent. Thus, it differs conceptually from the classical plane wave and may serve as a its regularization. We also give the polynomial solutions of the q-d'Alembert equation.

Characterization of 16-dimensional Hughes planes

The well-known finite Hughes planes have compact analoga with 16-dimensional point space. The automorphism group of such a plane is a 36-dimensional Lie group. Theorem: Assume that the compact projective plane $\cal P $ is not isomorphic to the classical Moufang plane over the octonions. Let $\Delta $ be a closed subgroup of $\hbox {Aut} \,\cal P $. If $\dim \Delta \ge 31$ and if $\Delta $ has a normal torus subgroup, then $\cal P $ is a Hughes plane, $\Delta = \hbox {Aut} \,\cal P $, and $\Delta ^{\prime } \cong \hbox {PSL}_3 \Bbb H $.

On generalized plane strain poroelasticity

Generalized plane strain is characterized by geometries of cylinders of finite or infinite length with boundary conditions that do not change in the direction of the generators. The resultant field generally contain three-dimensional stress and strain components, but has only two-dimensional spatial dependency. In addition to the classical plane strain deformation, it admits axial strain, warping, torsion, shear, bending, among other elementary deformation modes. The study of generalized plane strain solutions has so far been limited to elasticity. The present paper seeks the parallel development for poroelasticity. Due to the mathematical difficulty arising from the coupled nature of poroelasticity, analyses here are restricted to materials with orthotropic properties and cylindrical geometries whose longitudinal axis coincides with one of the material principal axes.

A case for superficial rhytidectomy

The first written accounts of the development of face-lift operations are credited to surgeons such as Miller, Kolle, and Bettman.le5 These early rejuvenation techniques were primarily designed to rid the patient of lateral facial wrinkling. Over the past two decades, the need for superficial musculoaponeurotic system (SMAS) manipulation during rhytidectomy has been widely accepted by most surgeons. In recent years, more complex aesthetic facial procedures, such as the deep-plane and composite rhytidectomies, have been popularized in the literature. Although numerous articles have attempted to show the benefits of these more complex, multilayered face-lift techniques, none definitively show a longer-lasting result.6-10 It is not the intent of this article to describe in detail the various rhytidectomy techniques in current use, but rather to present evidence supporting the superficial plane rhytidectomy with SMAS manipulation as the procedure of choice for facial rejuvenation. We believe it provides predictable, long-lasting results with minimal risks and a short recovery period. To avoid confusion in discussing this issue, the nomenclature used in rhytidectomy must be reviewed. Most authors categorize major rhytidectomy techniques into four generations (Table 1).7111 The subcutaneous rhytidectomy (first-generation) involves total subcutaneous undermining, while leaving the deep anatomic structures unchanged.5J2-14 Secondgeneration rhytidcctomies use various SMAS and platysmal suturing techniques (plication or imbrication) to reposition the facial soft tissues.11115-18 The deep-plane techniques (third generation) involve both platysmal and cheek fat repositioning; the classic planes of dissection are prezygomaticus in the cheek, subplatysmal in the lower face, and preplatysmal in the neck.6,7 The