Circle Plane Research Articles

Supercoiling transitions of closed DNA.

Conformations of closed DNA molecules are considered within a simple elastic model. The interplay of elastic energies of bending and twisting can lead to supercoiling induced by variations of the linking difference. These shape transformations are studied in the limit of a large length to thickness ratio of the molecule. Stationary shapes and energy diagrams are obtained by solving shape equations for closed rings. Four different families of stationary shapes can be distinguished: (i) planar circles, (ii) nonplanar rings, (iii) self-interacting rings, and (iv) interwound configurations. They all occur as shapes of minimal energy in the phase diagram of supercoiling. The transitions between the different regions in this phase diagram can be either continuous or discontinuous. The sequence of shape transitions turns out to be sensitive to the precise values of the elastic parameters. The buckling instability of the circle changes within the range of physically accessible values from continuous to discontinuous behavior.

The Twist, Writhe and Overall Shape of Supercoiled DNA Change During Counterion-induced Transition from a Loosely to a Tightly Interwound Superhelix: Possible Implications for DNA Structure in Vivo

A cryo-electron microscopy study of supercoiled DNA molecules freely suspended in cryo-vitrified buffer was combined with Monte Carlo simulations and gel electrophoretic analysis to investigate the role of intersegmental electrostatic repulsion in determining the shape of supercoiled DNA molecules. It is demonstrated here that a decrease of DNA-DNA repulsion by increasing concentrations of counterions causes a higher fraction of the linking number deficit to be partitioned into writhe. When counterions reach concentrations likely to be present under in vivo conditions, naturally supercoiled plasmids adopt a tightly interwound conformation. In these tightly supercoiled DNA molecules the opposing segments of interwound superhelix seem to directly contact each other. This form of supercoiling, where two DNA helices interact laterally, may represent an important functional state of DNA.In the particular case of supercoiled minicircles (178 bp) the ΔLk = -2 topoisomers undergo a sharp structural transition from almost planar circles in low salt buffers to strongly writhed "figure-eight" conformations in buffers containing neutralizing concentrations of counterions. Possible implications of this observed structural transition in DNA are discussed.

Models and theorems of the classical circle planes

Models and theorems of the classical circle planes

A direction finding ionosonde for ionospheric propagation research

A direction finding ionosonde consisting of a seven‐element, 50‐m crossed baseline interferometer has been developed to provide group path delay, azimuth, and elevation of each resolved mode over the sounding frequency range. Direction finding (df) ionograms have been acquired automatically, using various noncooperative compatible sounder transmissions. A diurnal sequence of df ionograms obtained over a 2344‐km mid‐latitude path is reported, including plots of group delay, azimuth, and elevation versus frequency, as well as an azimuth/elevation scatterplot showing the distribution of angular occupancy for the path. The results show modes arriving closest to the great circle bearing (that is, with lowest azimuth error) arrive via the lowest elevation angles and with least group delay. Greatest bearing accuracy was observed on daytime sporadic E modes; measured at 0.85 MOF, the azimuth standard deviation was 0.3°. One‐hop F region low‐ray azimuth measured at 0.85 MOF showed standard deviations between 0.4° and 0.98°. Fractional degrees per megahertz linear azimuth gradients with frequency were also observed at various times of day. One‐hop F region high rays showed smoothly varying elevation as a function of frequency, increasing with group path delay as expected. Variable lateral deviation from the great circle plane was observed for high rays up to 5° in azimuth equatorward at various times of day; azimuth standard deviation at 0.85 MOF varied from 0.91° to 1.7°.

Influence of Fluctuations on DNA Curvature: A Comparison of Flexible and Static Wedge Models of Intrinsically Bent DNA

Matrix-generator and Monte Carlo methods have been employed to study the influence of thermal fluctuations on the overall sizes and shapes of curved pieces of DNA. The DNA model involves the independent angular parameters relating successive base-pair steps: the sequence-dependent equilibrium values and fluctuations of the twist, tilt, and roll angles. The curved sequence under study is the (A 5X 5) n repeating polymer, the AA and XX steps having different equilibrium roll and twist values. Both planar circles and superhelices are analysed. Detailed comparison is made between the rigorous statistical mechanical representation of the DNA and simplified static models currently used in the literature. That is, a more realistic "flexible wedge" model is contrasted with the existing "static wedge" model of DNA curvature, which is demonstrated to be inadequate. The size of the coils is described by the unperturbed root-mean-square end-to-end distance and the shape by a ratio of the principal moments of the radius of gyration. The moment ratios indicate that when DNA is relatively short (e.g. its length is shorter than half a turn of the static superhelix), the flexible chains are more "short and thick" than the static structure. The end-to-end distances, however, are practically the same in the two models. For longer DNA fragments, the flexible chain is more extended in terms of the end-to-end distance and more globular in terms of the moment ratio. Thus, fluctuations "blur" the curvature of longer DNA fragments compared with static models. Furthermore, the overall average shape of slightly curved DNA subject to natural bending and twisting fluctuations is essentially indistinguishable from that of the corresponding "straight" DNA. Such configurational similarities are apparently responsible for the relative insensitivity of the polyacrylamide gel matrix to small degrees of DNA curvature. These findings raise serious questions regarding the quantitative estimation of wedge angles in DNA from electrophoretic experiments, based on static models. Comparison between planar circles and superhelices shows that when fluctuations are considered, the flexible circles are more spherical than the superhelices. The results imply that when DNA bending is exactly "in phase" with the helical repeat (i.e. when the DNA loop is exactly planar at 0 K), the DNA coil is packed more tightly than when bending and twisting are "out of phase" (and a superhelix is formed at 0 K). This finding is consistent with polyacrylamide gel electrophoresis data testifying to an increase in DNA retardation when twisting is more precisely "tuned". Finally, it is found that asymmetry of bending in the roll co-ordinate can produce significant macroscopic curvature of DNA. "Skewed" bending of AA steps in the A 5X 5 polymer is indistinguishable from symmetric bending with the same mean value and fluctuation of roll. Asymmetric rolling in the AA dimer thus can introduce substantial curvature in DNAs containing A n · T n blocks, even if the lowest energy stacked arrangement of AA base-pair steps is nearly flat, thereby resolving apparent contradictions between the crystallographic and solution structures of chains containing A n · T n tracts.

A note on graphs and sphere orders

AbstractA partially ordered set P is called a k‐sphere order if one can assign to each element a ∈ P a ball Ba in Rk so that a < b iff Ba ⊂ Bb. To a graph G = (V,E) associate a poset P(G) whose elements are the vertices and edges of G. We have v < e in P(G) exactly when v ∈ V, e ∈ E, and v is an end point of e. We show that P(G) is a 3‐sphere order for any graph G. It follows from E. R. Scheinerman [“A Note on Planar Graphs and Circle Orders,” SIAM Journal of Discrete Mathematics, Vol. 4 (1991), pp. 448–451] that the least k for which G embeds in Rk equals the least k for which P(G) is a k‐sphere order. For a simplicial complex K one can define P(K) by analogy to P(G) (namely, the face containment order). We prove that for each 2‐dimensional simplicial complex K, there exists a k so that P(K) is a k‐sphere order. © 1993 John Wiley & Sons, Inc.

Different aspects in designing mathematics education: Three examples from the Freudenthal Institute

Three teaching designs are offered. They were developed for 12 to 16 year old pupils taking into account their personal differences in terms of gender (in “Making Up the Cube”), culture (in “Mecca”), knowledge, and use of calculators in “The integrating calculator.” The designs concern the representation of three-dimensional objects in the plane, directions and great circles on the sphere, and exploration of the integrating calculator.

On the shape of a two-dimensional bubble in uniform motion

Consider a two-dimensional bubble moving with speed U through an unbounded, inviscid fluid. Let all lengths be normalized by T/ rhoU quot;SUP 2quot; where T is the surface tension. Then the shape of the bubble depends on a single parameter GAMMA=2 DELTAp/ rhoU quot;SUP 2quot; -1, where deltap=p quot;SUB bquot; -p quot;SUB infinityquot; is the difference between the bubble pressure and the ambient pressure. We obtain solutions for the bubble shape over the whole range of GAMMA-values that are physically relevant. The formulation involves a mapping from an auxiliary circle plane zeta where the flow field is known. The problem then reduces to solving an infinite set of nonlinear algebraic equations for the coefficients in the mapping function. To a first approximation, when GAMMA right arrow infinity, the bubble takes an elliptical shape of aspect ratio (1+ 2/3 GAMMA quot;SUP -1quot; )/(1-2/3 GAMMA quot;SUP -1quot; ) flattened in the flow direction. The solution correct to order GAMMA quot;SUP -5quot; is then obtained which is fairly accurate for GAMMA as low as 2. When GAMMA=0 the exact, nonlinear solution for the bubble shape is given by x= 1/3(1/3cos phi-1/27cos3 phi), y= 1/3(5/3sin phi+1/27sin3 phi). We can then obtain a perturbation solution for GAMMA right arrow0 correct to order GAMMA quot;SUP 6quot; . This solution, useful in the range 0.75 GAMMA-0.4537, even gives reasonable descriptions of non-convex bubble shapes for GAMMA0 down to the pinch-off limit GAMMA * when the bubble ceases to be simply connected. It is remarkable that a simple analytical representation correct to order GAMMA quot;SUP 2quot; analytically yields a value for GAMMA * of -0.4548, i.e. within 0.3% of the correct value; naturally, the higher-order approximations are even more accurate. While the present results eliminate the need for direct numerical computations over most of the range of GAMMA, such results, too, are presented. Finally, the dependence of the bubble geometrical parameters, Weber number and added mass on GAMMA is determined. (A)

Special circles in mechanics

W. Thomson and P. G. Tait (Principles of Mechanics and Dynamics, Dover, 1962) noted that for any deformation at any particle there is at least one plane circle of material elements all of which suffer precisely the same stretch. It has been shown that these material elements suffer no shear. These are typical of the results which may be obtained by considering ellipsoids associated with real non-singular second-order tensors. It is shown that the results may also be obtained provided the second-order tensor has at least one zero eigenvalue. Applications to strain, both finite and infinitesimal, rate of strain and stress are presented.

2-Type Integral Surfaces in $S^5(1)$

The main purpose of this paper is to classify integral surfaces of the unit sphere $S^5(1)$ which are mass-symmetric and of 2-type. If we consider $S^5(1)$ as a Sasakian manifold, then we prove that a mass-symmetric 2-type integral surface of $S^5(1)$ lies fully in $S^5(1)$ and is the product of a plane circle and a helix of order 4 or the product of two circles.

A Note on Planar Graphs and Circle Orders

A partially ordered set P is called a circle order if one can assign to each element $a \in P$ a circular disk in the plane $C_a $ so that $a < b$ and only if $C_a \subset C_b $. To a graph $G = ( V,E )$ associate a poset $P( G )$ whose elements are the vertices and edges of G. $v < e$ in $P( G )$ exactly when $v \in V, e \in E$, and v is an endpoint of e. It is shown that G is planar if and only if $P( G )$ is a circle order.

Spatial Price Competition: A Network Approach

Ohta 1988; Stahl 1989) and locational welfare (Stern 1972; Capozza and Van Order 1977; Benson 1984), we must not forget that the geographic properties of these models invariably have been naive. Besides employing several assumptions of parametric uniformity (for example, identical consumer demand and a constant transportation rate), researchers usually have simplified the geographic context for interfirm competition in two other important ways. First, the boundary or end firm problem has commonly been dealt with by envisaging an infinite (two-dimensional) plane (Liisch 1954; Mills and Lav 1964; Greenhut, Hwang, and Ohta 1975) or finite circle (Chamberlin 1953; Eaton 1976; Novshek 1980) over which competing firms locate. Only in a few cases, apparently, have spatial markets been depicted as both finite and bounded (Hotelling 1929; Eaton and Lipsey 1975; Carruthers 1981; Okabe and Suzuki 1987). As an extension of Eaton and Lipsey (19751, Okabe and Suzuki (1987) have discussed stability conditions and configuration of firms on a bounded two-dimensional space as a function of the number of spatial rivals, noting the instabilities associated with market boundaries. While we certainly agree with Lancaster (1979, p. 191) that it is a judgment call “as to how important events near the boundary are compared to events in the interior,” we feel that since the real world is indeed characterized by a multiplicity of (natural and man-made) boundaries, more effort must be made to accommodate the end (exterior) firm phenomenon in spatial competition models (Coyte et al. 1988; Hay and Johnston 1980; Fik 1989). Second, researchers often have assumed that the unbounded spatial market is in long-run equilibrium without ever establishing just how this equilibrium has been attained (recently, for example, Beckmann 1989; Capozza and Van Order 1989). In other words,

A local classification of 2-type surfaces in 𝑆³

The only spherical surfaces in E 4 {E^4} that are either of 1 1 -type or of 2 2 -type are portions of ordinary spheres, minimal surfaces in S 3 {S^3} , and Riemannian products of two plane circles of different radii.

Inviscid analysis of two-dimensional airfoils in unsteady motion using conformal mapping

Using a conformal mapping technique, a procedure for calculating the irrotational flow about a two-dimensional airfoil of arbitrary shape in unsteady motion is developed. The two-dimensional form with a sharp trailing edge is transformed into a unit circle by two successive transformations. The latter of the two is a modified version of the Gershgorin integral equation, which yields solutions much more accurate than those obtained with panel methods. The change in circulation around the airfoil due to unsteadiness is modelled by discrete vortices which are shed from the trailing edge and allowed to move freely with the local stream. The strength of these vortices is determined by the Kutta condition that, at each time step, the velocity be zero at the trailing edge in the circle plane. The procedure is efficient since the integral equation is solved only once. Numerical examples are presented for a sinusoidal heaving motion and for an impulsively started airfoil.

The Osculating Spiral

This article describes a method of approximating nonplanar space curves by cylindrical spirals rather than by planar circles. Readers familiar with osculating circle approximations should skip to the middle of the article. Readers unfamiliar with osculating circles will only need some familiarity with the geometry of vectors in three dimensions, especially the dot and cross products, along with some knowledge of parameterization of curves in space. One way to approximate a space curve by a more simple curve is to compare it with the so-called osculating circle. [1]

Earth structural patterns and rhythmic tectonism

Earth structural patterns and rhythmic tectonism

2-Type surfaces in S 3

It is proved that the Riemannian product of two plane circles of different radii are the only compact surfaces of the 3-spheres in R4 which can be constructed in R4 by using eigenfunctions associated with two different eigenvalues of their corresponding Laplacians.

Discrete vortex computation of separated airfoil flow

LOW-SPEED unsteady separated flow may be computed from Eulerian finite-difference or -element representations of the Navier-Stokes equations or by Lagrangian vortex tracking methods. The latter have been used to model both two-dimensional separated airfoil flows at high Reynolds numbers1'2 as well as numerous bluff-body flows.3'5 A major difficulty with the basic vortex method is that it is inviscid; hence, separation, except at sharp edges, must be predicted by some other means. Where empirical prediction is difficult, boundary-layer calculation (viscous-inviscid matching) has been used, but it is possible to predict laminar separation too early2 unless the discrete vortex representation is started upstream of the separation point. Viscous diffusion is usually modeled by vortex blob and random walk techniques.6'7 The results presented in this synoptic were computed using the inviscid cloud-in-cell vortex method.8 This is a mixed EulerianLagrangian method in which descrete vortices are tracked through a grid on which the velocity field is computed by a finite-difference method. It has the advantage of faster computation when large numbers of vortices are present but a poorer resolution of the velocity field. Contents Cloud-in-Cell Method In this method, the circulation of the vortices shed from the body is distributed by a weighting function (bilinear in the present calculations) onto a grid. The stream function \[/ is then computed from the Poisson equation V 2 ^=— w, linking \j/ to the resulting vorticity distribution co on the grid. This equation was expressed in central difference form and solved by a combination of fast Fourier transforms and Gaussian elimination. In order to avoid interpolated boundary conditions for \I/, the airfoil was transformed into a circle of radius R using the Joukowski transformation. The computation was carried out in the circle plane on a regular n x m polar mesh (rjtBk ) , where rj=R(\+2ir/m)J and Discrete vortices were shed into the flow from the two points on the circle corresponding to the trailing edge of the airfoil and a separation point just behind the leading edge. In the present study, the flow is over an 11%-thick Joukowski airfoil at 30-deg incidence, and the leading-edge separation point was fixed empirically at 1% chord, c, from observations of the separation point on a similar airfoil

Drift motions of small‐scale irregularities in the high‐latitude F region: An experimental comparison with plasma drift motions

On the evening of January 6, 1986, coordinated observations were carried out with the Johns Hopkins University Applied Physics Laboratory HF coherent scatter radar at Goose Bay, Labrador, and the SRI International incoherent scatter radar at Sondre Stromfjord, Greenland. The common field of view comprised a section of high‐latitude F region ionosphere centered on the great circle plane between the radar sites. Over a 40‐min period, the HF radar observed strong backscatter from small‐scale (13.9 m) field‐aligned irregularities. The bulk line‐of‐sight drift velocity of the irregularities is deduced from the backscatter data. The returns collected simultaneously with the incoherent scatter radar are processed for estimates of the mean line‐of‐sight ion velocity. Approximately 100 distinct comparisons are possible between the two sets of velocity estimates. Reversals exceeding 1000 m/s are present in both. In this paper, we demonstrate a correspondence between the measured irregularity and ion drifts that is consistent with the supposition that the motion of the irregularities is dominated by convective drift of the ambient plasma. This indicates that the small‐scale irregularities detected by HF radars in the high‐latitude F region can serve as tracers of ionospheric convective drift.

Computation of radiation impedance of stepped piston radiator

A computer program was devised for applying the Helmholz integral formulation to determine the pressure distribution on the surface of a stepped piston radiator, which consists essentially of a series of frustrums of cones and sectors of plane circles. The same program can also be used for various axially symmetrical modes of vibration of the sphere and of the plane piston by suitably specifying the values of the input parameters that determine the shape of the surface and the angle between the normal to each surface element and its direction of vibration. Good agreement is obtained (usually < 1% error) with the known analytic solutions for these latter radiators for ka⩽ 2, using only ten node from pole to equator. The complex radiation impedance as a function of ka was computed for Lubell Laboratories model 98 underwater loudspeaker that vibrates by separating along its flexible equator. The results are compared with those of a sphere that vibrates likewise. [This work was supported by Lubell Laboratorie...