Secant Plane Research Articles

Secant planes of a general curve via degenerations

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite.

О круговых сечениях поверхности второго порядка

A brief overview of the history of conic sections is given. Circular sections of ellipsoids and hyperboloids with planes passing through the center of the surface are considered. In general, there are two such secant planes. Generalizing the concept that arose in rigid-body mechanics, a straight line passing through the center of an ellipsoid is called the Galois axis if the orthogonal plane intersects this ellipsoid along a circle. Let us consider the pencil of planes passing through the intermediate principal axis of a triaxial ellipsoid. Each section of an ellipsoid with such a plane is an ellipse, one of the axes of which coincides with the intermediate principal axis of the ellipsoid. When the secant plane rotates around the intermediate principal axis of the ellipsoid, the length of the other axis of the ellipse continuously changes, taking values between the lengths of the minor and major axes of the ellipsoid. Therefore, some such section is a circle whose diameter is the intermediate principal axis of the ellipsoid. A triaxial ellipsoid has two such sections. They transform into each other when mirrored relative to the plane passing through the intermediate and other principal axes of the ellipsoid. Both Galois axes are orthogonal to the intermediate principal axis of the triaxial ellipsoid, and for a non-sphere ellipsoid of rotation, both Galois axes coincide with one axis and are orthogonal to the other principal axes of the ellipsoid. A method for constructing Galois axes from the known principal axes of an ellipsoid is proposed. This construction serves as one of the natural examples of geometric problems. In addition, the Galois axis can be correctly defined not only for the ellipsoid (for which it was originally introduced), but also for some other classes of centrally symmetric surfaces, including hyperboloids.

CONSTRUCTION OF SECTIONS OF POLYHEDRON METHOD OF TRACES

In this article it is proved that in solving problems on the construction of sections of polyhedra, students not only perform constructions, they also apply axioms, properties of planimetry and stereometry, but also learn algorithmic thinking, the ability to reason logically, make correct arguments and conclusions. It is established that the solution of problems on the construction of sections of polyhedra occupies a special place in the process of forming a spatial representation and in the development of mathematical thinking, both students and schoolchildren. Based on the definition of the trace of the secant plane, the rules for constructing sections of the polyhedron by the traces method are formulated. Problems on construction of sections of polyhedra are developed in the case when:: the section of the prism is given by the trace l , which is located on the plane of the base of the prism and does not have common points with the base of this prism and by point K , belonging to some side rib; the secant plane is defined by the trace l and some point M , belonging to the side rib of the pyramid; the section of the pyramid is determined by points M, N, K two of them are located on different ribs, and the third is the internal point of the face of this pyramid.

Characterization of the chaos-hyperchaos transition based on return times.

We discuss the problem of the detection of hyperchaotic oscillations in coupled nonlinear systems when the available information about this complex dynamical regime is very limited. We demonstrate the ability of diagnosing the chaos-hyperchaos transition from return times into a Poincaré section and show that an appropriate selection of the secant plane allows a correct estimation of two positive Lyapunov exponents (LEs) from even a single sequence of return times. We propose a generalized approach for extracting dynamics from point processes that allows avoiding spurious identification of the dynamical regime caused by artifacts. The estimated LEs are nearly close to their expected values if the second positive LE is essentially different from the largest one. If both exponents become nearly close, an underestimation of the second LE may be obtained. Nevertheless, distinctions between chaotic and hyperchaotic regimes are clearly possible.

Hyperovals on [formula omitted] left invariant by a group of order [formula omitted

Three new infinite families of hyperovals on the generalized quadrangle H(3,q2) (q=ph, p a prime) of sizes 6(q+1), 12(q+1)2 (if q>7) and 6(q+1)2 (if p>3) are constructed. Furthermore they turn out to be invariant under the action of a linear collineation group of order 6(q+1)3 that fixes no point or line in a secant plane of H(3,q2). In particular the hyperovals of size 6(q+1)2 are transitive.

A combinatorial characterization of parabolic quadrics

In this paper we prove that, in PG(3, q), q odd, a (q 2 + q +1)-set of type ((n − 1)q + 1, nq + 1, (n + 1)q + 1)2, 1 ≤ n ≤ q, and type (a, b, c, d)1, with , such that any external line is contained in exactly one ((n − 1)q + 1)-secant plane is a parabolic quadric.

Model approach to restoration of salinity data: Evidence from the Sea of Azov

During the study of hydrophysical processes, thermohaline measurements are usually carried out at a limited number of points. Thus, determination of seawater salinity at an arbitrary point of the reservoir on the basis of observed values is an interesting problem from the scientific point of view. Below, we describe the solution of this problem using mathematical modeling. The mathematical model of processes of matter transport and spreading in water consists of two components: a hydrodynamic model of the reservoir and a model describing convective-diffusive transport of a substance. The shallow reservoir of the Sea of Azov (up to 14 m), commonly accepted as a basin with a flat even bottom, is of a certain scientific interest. Actually, new large-scale charts demonstrate strong differentiation of the bottom topography in the coastal shelf and open sea regions [2]. Equations for shallow water are usually used for the calculation of currents in shallow basins, while currents in the deep basins are modeled by 3D equations of fluid motion. If a reservoir has large, shallow and sufficiently deep regions, application of shallow water equations will not yield a reliable pattern of currents in the deep-water part of the basin. At the same time, application of 3D equations for the entire basin can require the use of curvilinear grids (at least over the vertical) or preliminary transformation of the initial irregular domain into a regular one. Such transformations strongly complicate the formulation of the problem and its numerical solution. However, if we identify domains of shallow and deep zones of the basin, it is possible to apply the shallow-water equations and 3D equations of fluid motion without preliminary transformation of the calculation area. In this case, it is possible to use the finite difference methods using uniform rectangular grids, which undoubtedly simplifies the solution of the problem. The advantage of the model applied here is its simple mathematical description and easy numerical realization especially using the algorithms of parallel computation. The initial 3D modeling domain is the water medium of the basin limited from above and below by the sea surface and seafloor, respectively. Let us locate a horizontal secant plane P at a distance from the unperturbed surface of the basin P 0 at a depth equal to the maximal depth of the shallow region and divide the entire modeling domain into the upper (I) and lower (II) layers (Fig. 1). Layer I includes the entire coastal zone and the upper layer of the open sea. Layer II is the lower layer of the water column. The motion in layer I is described by the shallow-water equations:

Liquid–liquid phase separation in sodium borosilicate glass: Ordering phenomena in particle arrangement

A structure of 13.9Na 2O · 36.0B 2O 3 · 50.1SiO 2 glass heat-treated at 610 °C for 5 h (glass G1) and 10 h (glass G2) was studied by stereological method on the basis of electron microscopy data obtained from plane sections of samples. It has been found that liquid phase separation in the course of heat treatment leads to formation of spherical particles with the volume fractions 0.047 (G1) and 0.065 (G2). It is argued that the stage of growth takes place in the glass G1 and the Ostwald ripening stage occurs in the glass G2. The analyses of pair correlation functions of particle traces on a secant plane indicate elements of ordering in mutual arrangement of particles for the both glasses G1 and G2. This conclusion is confirmed by results of numerical simulation. The experimental data on the pair correlation functions are in agreement with the results of calculations presented in the literature. It is noted that the ordering elements in mutual arrangement of particles play a key role in light scattering and other physical properties of material.

Return time dynamics in dependence to choice of the poincare section

In this paper we study how different measures of chaotic dynamics estimated from return time sequences are sensitive to choice оf the Poincare section. We show that dynamical characteristics оf chaotic oscillations аrе less sensitive to displacements оf the secant plane than metrical and scaling characteristics of evolution dynamics.

Return times dynamics: role of the Poincaré section in numerical analysis

We study how different measures estimated from return time sequences are sensitive to choice of the Poincaré section in the case of chaotic dynamics. We show that scaling characteristics of point processes are highly dependent on the secant plane. We focus on dynamical properties of a chaotic regime being more stable to displacements of the section than metrical characteristics.

Ovoids of PG(3,q ), q Even, with a Conic Section

It is shown that if a plane of PG(3,q), q even, meets an ovoid in a conic, then the ovoid must be an elliptic quadric. This is proved by using the generalized quadrangles T2([Cscr ]) ([Cscr ] a conic), W(q) and the isomorphism between them to show that every secant plane section of the ovoid must be a conic. The result then follows from a well-known theorem of Barlotti.

On the problem of polydispersity of particles formed in the course of phase separation in a glass in determination of their pair correlation function by the stereological method

The integral equation relating the pair correlation function (PCF) of an ensemble of polydisperse spherical particles and the PCF of their traces on the secant plane has been derived. This equation can be used in discussing the problem concerning the determination of the PCF of particles formed in the course of phase transformation in a glass-forming melt from experimental data obtained using the plane section of a sample.

Extracting dynamics from threshold-crossing interspike intervals: Possibilities and limitations

In this paper we estimate dynamical characteristics of chaotic attractors from sequences of threshold-crossing interspike intervals, and study how the choice of the threshold level (which sets the equation of a secant plane) influences the results of the numerical computations. Under quite general conditions we show that the largest Lyapunov exponent can be estimated from a series of return times to the secant plane, even in the case when some of the loops of the phase space trajectory fail to cross this plane.

Analytical solution to the equation for pair correlation function of particles formed in the course of phase separation in a glass

The problem of determining the pair correlation function of phase-separated particles from the experimental data obtained for the plane section of a sample has already been consul and in the literature. The integral equation relating the pair correlation function of monodisperse spherical particulers and the pair correlation function of their traces on the secant plane was obtained, and the method of numerical solution of this equation was elaborated with the aim of determining the pair correlation function of places. The present paper describes the analytical solution to this equation.

Computing largest Lyapunov exponent from a sequence of return times: possibilities and limitations

We analyze whether it is possible to estimate dynamical characteristics of a chaotic attractor from а sequence of return times and study how the choice of a secant plane influences the result of reconstruction. Using Rossler model as аn example it is shown that the largest Lyapunov exponent can be determined from a series of return times even in that case, when not all the phase trajectories cross а secant plane.

Symbolic dynamics behind the singular continuous power spectra of continuous flows

We investigate a class of dissipative flows which possess singular continuous (fractal) power spectra. The attractor for these flows contains a saddle point, so that the times between the returns of an orbit onto a secant plane are unbounded. As a result, the Poincaré map does not reflect precisely the qualitative dynamics of the underlying flow: the power spectrum of this map can be discrete. We introduce the construction which takes into account that the time intervals between the intersections of the Poincaré plane can vary and construct a symbolic representation for the continuous process with logarithmic divergence in the distribution of such intervals. Spectral properties of the resulting symbolic sequences are shown to reproduce rather well those of the original flow.

Ovals in Translation Hyperovals and Ovoids

LetO; be an ovoid of PG(3,q),qeven, such that each secant plane section is an oval contained in a translation hyperoval. ThenO; is shown to be either an elliptic quadric or a Tits ovoid.

Classification of ovoids inPG(3, 32)

A classification of the ovoids inPG(3, 32) is completed with the aid of a computer. The ovoids are examined in terms of which ovals can possibly appear as secant plane sections. A weak necessary condition for two ovals to appear together as plane sections of an ovoid surprisingly turns out to be sufficient to demonstrate that the only possible secant plane sections are translation ovals. A known result regarding ovoids with such plane sections then identifies the ovoids as either elliptic quadrics or Tits ovoids.

The onset of stochastic pulsations at the early non-linear stage of the development of perturbations in the jet of an incompressible fluid propagating along a wall

The Korteweg-De Vries equation, which describes the non-linear propagation of perturbations in a jet of incompressible fluid emanating from a slit in a planar screen and propagating along a wall is considered. When account is taken of the natural vibrations of the wall, the equation becomes inhomogeneous. If an external action is specified in the form of a running wave, the particular solution of the inhomogeneous equation may be sought in an analogous form. As a result, the simplest problem in the theory of dynamical systems in the Hamiltonian formulation arises. As usual, the existence of a homoclinic structure in the neighbourhood of the separatrices is deduced from an analysis of a Poincaré transformation. Among the trajectories belonging to the homoclinic structure in the secant plane, there are some with properties which are formulated in terms of determinate chaos. A fundamentally important conclusion concerning the dual role of solitons at the non-linear stage of the wave motion of the fluid follows: on the one hand, they serve as the nuclei of large-scale coherent structures and, on the other hand, they are responsible for the onset of stochastic pulsations.

STEREOLOGICAL ESTIMATION OF TOTAL FIBER LENGTH IN ANISOTROPIC FIBER SUSPENSIONS

It is shown that the total fiber length in anisotropic fiber assemblies can be determined by stereological procedures. The main result obtained is the equation:where Λ is the total fiber length per unit volume of the assembly and v(_??_, Φ) is the number of crossings of the fibers and a secant plane of unit area whose normal is in the direction of (_??_, Φ).It is also shown that the reduced forms of the equation into two special assemblies are identical to the corresponding equations previously proposed by Saltykov.