Elliptic Lines Research Articles

Two kinds of real lines on real del Pezzo surfaces of degree 1

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, \({\text {Pin}}^-\)-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the difference between the total numbers of hyperbolic and elliptic lines is always equal to 16.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field$k$, generalizing the counts that over${\mathbf {C}}$there are$27$lines, and over${\mathbf {R}}$the number of hyperbolic lines minus the number of elliptic lines is$3$. In general, the lines are defined over a field extension$L$and have an associated arithmetic type$\alpha$in$L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group$\operatorname {GW}(k)$of$k$,\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]where$\operatorname {Tr}_{L/k}$denotes the trace$\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over${\mathbf {C}}$and${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in$\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Physical topology of three-dimensional unsteady flows with spheroidal invariant surfaces.

Scope is the response of Lagrangian flow topologies of three-dimensional time-periodic flows consisting of spheroidal invariant surfaces to perturbation. Such invariant surfaces generically accommodate nonintegrable Hamiltonian dynamics and, in consequence, intrasurface topologies composed of islands and chaotic seas. Computational studies predict a response to arbitrary perturbation that is dramatically different from the classical case of toroidal invariant surfaces: said islands and chaotic seas evolve into tubes and shells, respectively, that merge into "tube-and-shell" structures consisting of two shells connected via (a) tube(s) by a mechanism termed "resonance-induced merger" (RIM). This paper provides conclusive experimental proof of RIM and advances the corresponding structures as the physical topology of realistic flows with spheroidal invariant surfaces; the underlying unperturbed state is a singular limit that exists only for ideal conditions and cannot be achieved in a physical experiment. This paper furthermore expands existing theory on certain instances of RIM to a comprehensive theory (supported by experiments) that explains all observed instances of this phenomenon. This theory reveals that RIM ensues from perturbed periodic lines via three possible scenarios: truncation of tubes by (i) manifolds of isolated periodic points emerging near elliptic lines or by either (ii) local or (iii) global segmentation of periodic lines into elliptic and hyperbolic parts. The RIM scenario for local segmentation includes a perturbation-induced change from elliptic to hyperbolic dynamics near degenerate points on entirely elliptic lines (denoted "virtual local segmentation"). This theory furthermore demonstrates that RIM indeed accomplishes tube-shell merger by exposing the existence of invariant surfaces that smoothly extend from the tubes into the chaotic shells. These phenomena set the response to perturbation-and physical topology-of flows with spheroidal invariant surfaces fundamentally apart from flows with toroidal invariant surfaces. Its entirely kinematic nature and reliance solely on continuity and solenoidality of the velocity field render the comprehensive theory and its findings universal and generically applicable for (arbitrary perturbation of) basically any incompressible flow-in fact any smooth solenoidal vector field-accommodating spheroidal invariant surfaces.

Non-stationary vortex streets in shear flows

Spatially periodic vortex systems that form due to unstable shear flows are called vortex streets. A number of exact and asymptotic solutions of two-dimensional hydrodynamic equations describing nonstationary vortex streets have been constructed. It is shown that the superposition of the flow with a constant horizontal shear and its perturbations in the form of a nonmodal wave provides an exact solution that describes a nonstationary vortex street with rotating elliptic current lines. The width of the zone occupied by such a vortex street has been determined. The equation of separatrix separating vortex cells with closed current lines from an external meandering flow has been obtained. The influence of the quasi-two-dimensional compressibility and beta effect on the dynamics of vortex streets has been studied based on the potential vorticity transport equation. The solutions describing the formation of vortex streets during the development of an unstable zonal periodic flow and a free shear layer have been constructed using a longwave approximation.

Nonstationary Vortex Streets in Shear Flows

Spatially periodic vortex systems that form due to unstable shear flows are called vortex streets. A number of exact and asymptotic solutions of two-dimensional hydrodynamic equations describing nonstationary vortex streets have been constructed. It is shown that the superposition of the flow with a constant horizontal shear and its perturbations in the form of a nonmodal wave provides an exact solution that describes a nonstationary vortex street with rotating elliptic current lines. The width of the zone occupied by such a vortex street has been determined. The equation of separatrix separating vortex cells with closed current lines from an external meandering flow has been obtained. The influence of the quasi-two-dimensional compressibility and beta effect on the dynamics of vortex streets has been studied based on the potential vorticity transport equation. The solutions describing the formation of vortex streets during the development of an unstable zonal periodic flow and a free shear layer have been constructed using a longwave approximation.

Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

Cooperative Processing of Measurements in Pseudorange Radio-technical Finding System

Background/Objectives: The relevance of the current research is determined by the intensive development of multilateration systems which are widely used in time-positioning support systems. The authors propose to use the cooperative processing of coordinate information. Methods: In the present article a procedure of obtaining the ranging position lines with regard to each station is studied using a theory of redundant measurements which enables to implement cooperative processing. It is shown that using the least squares method, one can obtain the estimation of the required range-measurement parameters containing a minimum value of mean-square error. Findings: In our opinion, the new results of the study show that a multilateration system with one query station which forms a range line of position and with cooperative processing of Time Difference Of Arrival (TDOA) and Time Of Arrival (TOA), in some cases it is characterized by the improved accuracy compared to hyperbolic and elliptic systems. Measurement processing method studied in the present article enables to expand a system work area with respect to the accuracy for multilateration systems with short length base (distances between receiver stations). The offered procedures of radar data processing allow for a formation of range position lines that are indirectly obtained based on hyperbolic and elliptic lines of position. Cooperative processing of measurements in TOA and TDOA finding systems enables to improve the accuracy of the indirect range measurement. TDOA and TOA measurements do not improve the accuracy of the indirect range estimation with the offered method of measurement processing since the accuracy remains constant regardless of the number of stations. Application/ Improvements: The study results obtained could be used to upgrade the existing systems of aircraft position estimation and to make reasonable technical demands on the systems under development.

Gromov–Witten theory and cycle-valued modular forms

Abstract In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) ⁢ GL ⁢ ( H ) {\mathcal{L}^{(2)}_{+}{\rm GL}(H)} action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight ( 3 , 3 , 3 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

Structural Analysis of 1000MW Ultra Supercritical Boiler’s Steam-Water Separator

The steam-water separator is the most important pressure part of the Ultra Supercritical Boiler（USB）. The steam-water separator’s service life is related to the monitoring of the security issue of the UCB. The steam-water separator typical structures of the domestic 1000 MW ultra-supercritical boiler are compared. The difficult problems in the separator’s manufacturing process are analyzed. The separator coupling stress field is simulated through the finite element method. The results show that the maximal stress locates on the long shaft position of the elliptic hole interfingering lines where has the serious stress concentration. That point is the key spot of the strength analysis and life monitoring during the boiler operation process.

Intrinsic signs and lower bounds in real algebraic geometry

Abstract. A classical result due to Segre states that on a real cubic surface in ℙ ℝ 3 $\mathbb {P}^3_\mathbb {R}$ there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2 m - 3 $2m-3$ in ℙ ℝ m $\mathbb {P}^m_\mathbb {R}$ .

Approximation of characteristic parameters for elliptic and circular microshield lines using a robust fuzzy approach

We propose a fuzzy learning method by a descent technique, to calculate the characteristic parameters of elliptic- and circular-shaped microshield lines.From input–output data, generated using conformal mapping technique, the inference rules expressing the input output relation of the data are obtained automatically using the proposed fuzzy inference system. The learning speed of this method is substantially higher than that of the neural model. The numerical results are in very good agreement with the results reported elsewhere. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:2648–2653, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26335

Harmonic oscillator under Lévy noise: Unexpected properties in the phase space

A harmonic oscillator under the influence of noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general Lévy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial.

The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines

The purpose of this paper is a description of a model of Kummer surfaces in characteristic 2, together with the associated formulas for the pseudo-group law. Since the classical model has bad reduction, a renormalization of the parameters is required, that can be justified using the theory of algebraic theta functions. The formulas that are obtained are very efficient and may be useful in cryptographic applications. We also show that applying the same strategy to elliptic curves gives Montgomery-like formulas in odd characteristic that are faster than the classical ones, and we recover already known formulas by Stam in characteristic 2.

NEURAL MODELS FOR THE ELLIPTIC- AND CIRCULAR-SHAPED MICROSHIELD LINES

This article presents a new approach based on artificial neural networks (ANNs) to calculate the characteristic parameters of elliptic and circular-shaped microshield lines. Six learning algorithms, bayesian regularization (BR), Levenberg-Marquardt (LM), quasi- Newton (QN), scaled conjugate gradient (SCG), resilient propagation (RP), and conjugate gradient of Fletcher-Reeves (CGF), are used to train the ANNs. The neural results are in very good agreement with the results reported elsewhere. When the performances of neural models are compared with each other, the best and worst results are obtained from the ANNs trained by the BR and CGF algorithms, respectively.

The First Quarto of "Romeo and Juliet" (review)

Reviewed by: The First Quarto of “Romeo and Juliet.” Beatrice Groves William Shakespeare. The First Quarto of “Romeo and Juliet.” Ed. Lukas Erne. Cambridge: Cambridge University Press, 2007. Pp. xvii, +156 +. appendices. $85.00. This is an excellent edition of a wonderful play. Lukas Erne's annotations celebrate Q1 for what it is, rather than simply as an interesting adjunct to the received text. The challenging introduction, careful collations, insightful notes, and useful appendices open up the play to the wider audience it deserves, and the edition marks a distinct step forward from Cambridge's previous early quarto editions by giving textual notes at the bottom of each page. The appendices on the use of Q1 in an Elizabethan literary anthology, and detailing the 199 readings from Q1 adopted by eighteenth-century editors, show how the play—almost hidden in the modern theatrical tradition—has influenced readers and editors throughout its history. Erne's edition is perhaps at its best in its refreshingly open acknowledgment, and even celebration, of the fact that there is no single, simple explanation for the distinctiveness of Q1. As Erne argues, the text shows the influence of memory, while clearly not being simply the result of "memorial reconstruction"; passages show evidence of cutting, although this is not simply a text that has been shortened for a provincial tour; sections have been carefully revised, although Q1 is not simply an authorial first draft. Erne's edition will encourage academics and students alike to embrace the complexity of genesis to which this quarto witnesses, providing as it does an insight into the workings of Elizabethan theater and print culture, and into the creative mind of the revising author. Erne's most distinctive, and provocative, emphasis is on the possibility that some of the differences between the two texts are due to Q2's being intended for reading, Q1 for performance. He extrapolates from the fact that Q2 "establishes verbally what Q1 suggests visually" (note to 17.27 SD 2) the idea that Q1 eliminates "what the reader needs to be told but a spectator would be about to see" (note to 8.8). Erne's interest in the literary aspect of Shakespeare's texts, however, can lead him into dangerous waters—such as suggesting that the "roundness" of Q2's characterization is related to a readerly rather than theatrical reception, the logical extension of which would be that two-dimensional characterization is typical of Shakespearean theater (33–34). More often, however, Erne is at pains to show that comparisons between the two texts need not lead to the detriment of Q1—as Edward Capell expressed it, "here and there a various reading that does honour to the Poet's judgement" (170). In many of the single-word variants—such as Q1's "flattering eye [Q2: truth] of sleep" (18.1)—it is the earlier text that has more immediate appeal. Mercutio's Queen Mab speech, likewise, can be persuasively argued to be superior in Q1 and, to the sympathetic ear, elliptic lines (such as 10.122) begin to [End Page 525] sound like the poetic compression of the author of the sonnets, rather than garbled misremembering. Other effective differences are found in what appear to be cuts: Q1 ends the scene between Juliet and the Friar with the words "Be sure thou send for my dear Romeo" (15.92), as opposed to the verbosity of the Friar in Q2. While this is not a complex dramatic technique, leaving the doom-laden words hanging in the air suggests theatrically sensitive redaction that may actually produce a superior dramatic moment. Erne's concentration on Q1 means that he works hard to find as acceptable readings that editors have previously denigrated. A particularly attractive example lies in Mercutio's roll call of abusive names for his friend—"Romeo! Madman! Humours! Passion! Liver!" (5.8). Q2 ends with "Lover!" but the substitution of a letter may not simply be a mistake. Erne notes that the liver was the ancient seat of love and, while the Q2 reading must remain the most likely one, there is something appealingly abusive about "Liver!" that creates a funnier line, and as such, may take us...

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration $${\cal R}$$ (q) on the set $${\cal L}$$ (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $${\cal R}$$ +(q) and $${\cal R}$$ ?(q) of $${\cal R}$$ (q) to the set $${\cal L}$$ +(q) of secant (hyperbolic) lines and to the set $${\cal L}$$ ?(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $${\cal R}$$ ?(q) is pseudocyclic. We further show that the coherent configurations $${\cal R}$$ (q 2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $${\cal R}$$ +(q 2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $${\cal R}$$ +(q 2) and $${\cal R}$$ ?(q 2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.

Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence

We derive analytic criteria for the existence of hyperbolic (attracting or repelling), elliptic, and parabolic material lines in two-dimensional turbulence. The criteria use a frame-independent Eulerian partition of the physical space that is based on the sign definiteness of the strain acceleration tensor over directions of zero strain. For Navier–Stokes flows, our hyperbolicity criterion can be reformulated in terms of strain, vorticity, pressure, viscous and body forces. The special material lines we identify allow us to locate different kinds of material structures that enhance or suppress finite-time turbulent mixing: stretching and folding lines, Lagrangian vortex cores, and shear jets. We illustrate the use of our criteria on simulations of two-dimensional barotropic turbulence.

Three-dimensional theory of water impact. Part 1. Inverse Wagner problem

The three-dimensional problem of blunt-body impact onto the free surface of an ideal incompressible liquid is considered within the Wagner theory. The theory is formally valid during an initial stage of the impact. The problem has been extensively studied in both two-dimensional and axisymmetric cases. However, there are no exact truly three-dimensional solutions of the problem even within the Wagner theory. At present, three-dimensional effects in impact problems are mainly handled approximately by using a sequence of two-dimensional solutions and/or aspect-ratio correction factor. In this paper we present exact analytical rather than approximate solutions to the three-dimensional Wagner problem. The solutions are obtained by the inverse method. In this method the body velocity and the projection on the horizontal plane of the contact line between the liquid free surface and the surface of the entering body are assumed to be given at any time instant. The shape of the impacting body is determined from the Wagner condition. It is proved that an elliptic paraboloid entering calm water at a constant velocity has an elliptic contact line with the free surface. Most of the results are presented for elliptic contact lines, for which analytical solutions of the inverse Wagner problem are available. The results obtained can be helpful in testing other numerical approaches and studying the influence of three-dimensional effects on the liquid flow and the hydrodynamic loads.

On constructing kinematically admissible velocity fields in cold sheet rolling

The main objective of the presented study is to introduce a rigorous, systematic method for constructing kinematically admissible velocity fields, which are necessary in the analysis of the plastic forming processes by the upper bound method. The mathematical structure of the method is discussed in full detail. To demonstrate the application of the presented method, three different deformation models for the symmetric strip rolling are obtained. And also, to prove the superb accuracy of the deformation models obtained by the presented method, an upper bound solution for the deformation model subjected to the elliptic flow lines family is given. The results obtained in this analysis are compared with the results of modelling in [B. Avitzur, Metal Forming: Processes and Analysis, McGraw-Hill, New York, 1968], and the advantages provided by the presented method, and some new research directions are pointed out.

Two-dimensional elastic defects in orthotropic bicrystals

A method is presented for obtaining the closed form elastic solutions of two-dimensional inclusion and dislocation problems in orthotropic bicrystals. The bicrystal consists of two semi-infinite single crystals with a planar interface. The interface is a plane of symmetry for both crystals. The two single crystals are either perfectly bonded together or in frictionless contact with each other. Both inplane and antiplane deformations that include plane strain and plain stress conditions are considered. The method is based on Eshelby's method for inclusion problems in infinite isotropic solids. The analytic results for point forces, inclusions and dislocations are given. The elastic fields of the defects are represented by quasi-harmonic potentials that are associated with the defects. Explicit solutions for elliptic inclusions and dislocation lines are given as examples. Results for elliptic inclusions and dislocation lines of different orientations in cubic structures are also presented.