Showing papers on "Plane curve published in 1978"

On normal two-dimensional double point singularities

Abstract: Letq be a plane curve singularity and letp be the corresponding normal two-dimensional double point singularity. Let Γ and Γ1 be the topological types of the minimal and of the canonical resolutions ofp respectively. An algorithm is given for finding the equisingular type ofq in terms of Γ1. An algorithm is also given for finding all Γ1 corresponding to a given Γ. There is at most one such Γ1 in case Γ has no 1-cycles.

Aspheric grating for extreme ultraviolet astronomy

Abstract: A family of plane curves is developed which can diffract incident parallel rays to a point focus. These curves, termed diffractoidal curves, are rotated around an axis to produce surfaces of revolution correspondingly termed diffractoids, whose imaging properties for sources at infinity are studied by ray tracing in a few examples. The paraboloid emerges as a limiting case of the diffractoid. A comparison is made between the stigmatic focusing properties of the diffractoid and the toroidal grating.

Topological Hjelmslev planes

Abstract: An affine or projective Hjelmslev plane (henceforth A.H. or P.H. plane) is a generalization of an ordinary affine or projective plane, where two lines (points) may intersect in (be joined by) more than one point (line). Two points are neighbours if they possess more than one joining line, and two lines are neighbours if each point of either line has a neighbour on the other. These neighbour relations define equivalence relations on the set of points and the set of lines respectively. Their corresponding quotient spaces generate an ordinary plane called the quotient plane of the Hjelmslev plane. A projective (affine) plane is a topological plane if the set of points and the set of lines are Hausdorff spaces, and the operations of join and intersection (and parallelism) are continuous. The purpose of this paper is to investigate analogous topological notions in Hjelmslev planes. Our ideas are naturally highly motivated by the work of Salzmann (cf. [11] and [12]). Every ordinary affine plane has a unique projective extension. This result has a topological analogue, provided the plane is Desarguesian or locally compact and connected (cf. [11], §14 and [12], 7.15). However, for H-planes, the author and N. D. Lane have constructed in [8] a Desarguesian A.H. plane with no Desarguesian projective extension, and in [3] Drake constructed an A.H. plane with no projective extension. Hence, we shall first study topological A.H. planes, in Sections 1 to 6, and then consider topological P.H. planes in Sections 7 and 8. Examples of such planes are constructed in Section 9. A P.H. (A.H.) plane is topological if its point and line sets are topological spaces, the neighbour relations are closed, and join and intersection (and parallelism) are continuous. Our main objectives are, to determine when the quotient plane, endowed with the usual quotient topologies, is a topological plane, and to construct examples of topological H-planes over certain topological rings. In Section 3 we determined relationships between point and line topologies and use these to prove join and intersection are open maps. Section 4 contains the result that the quotient plane is topological if and only if the quotient map is open on the points. If the plane is locally compact and connected or a translation plane, then the quotient plane is shown, in Section 5, to be * The author gratefully acknowledges the support of the National Research Council of Canada.

Diffraction by two parallel slits in a plane

Abstract: The diffraction of scalar waves by two parallel slits in a plane screen is examined by a method which uses orthogonal functions and Fourier transformations. A solution is obtained in the case of a perfectly soft screen. Numerical results of the plane wave transmission coefficients for normal incidence are given for k=0.2∼4.0 and D=3∼10, where k is the wavenumber and D is the distance between slits.

The local Hilbert function of a pair of plane curves

Abstract: Let R be the local ring of a pair of plane curves at a point. In this paper it is proved that the Hubert function of such a ring changes by at most one at each stage, and it is essentially nonincreasing. 1. Introduction. Let 0 be an equicharacteristic two dimensional regular local ring and / and g be two elements of 0 which generate an ideal mn when/and g are tangential, i.e., when their initial forms have a common factor? Question 2. Is \Hi+x(R) — Hi(R)\ < 1 for all nonnegative integers il These questions are answered in the affirmative and the following theorem is proved.

Universal solutions in the hydrostatics of nematic liquid crystals

Abstract: Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 1. Background Material . . . . . . . . . . . . . . . . . . . . . . . . . 255 2. The Case when ~ Does not Vanish . . . . . . . . . . . . . . . . . . . 258 3. The Case Q~=0, when the s-Lines are Rectilinear . . . . . . . . . . . . . 260 4. The Case Ms=O, when the s-Lines are Plane Curves with K ~ 0 and curl b=O . 263 5. The Case ~ s =0 , when the s-Lines are Plane Curves and • and curl b are Different from Zero, Part I . . . . . . . . . . . . . . . . . . . . . . . 264 6. The Case s when the s-Lines are Plane Curves and • and curl b are Different from Zero, Part II . . . . . . . . . . . . . . . . . . . . . . 272 7. The Case ~ = 0 , when the s-Lines are neither Rectilinear nor Plane Curves. Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 274 8. Preliminary Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 278

Topology of curves and surfaces, and special topics in the theory of algebraic varieties

Abstract: Oscar Zariski's earliest papers originally appeared in 1924 and have been followed by a steady accretion ever since. That at least four volumes are required to publish his collected papers is an index to his productiveness and persistence; that they have been collected at all is a tribute to their continuing importance in the field of algebraic geometry.The first two volumes of Zariski's papers were published in 1973. Volume I, "Foundations of Algebraic Geometry and Resolution of Singularities, " was edited by H. Hironaka and D. Mumford, and Volume II, "Holomorphic Functions and Linear Systems, " was edited by M. Artin and D. Mumford.The papers contained in this third volume were originally published between 1925 and 1966, but the heart of the book is a sequence of papers, topological in nature, that appeared during the period 1928-1937. Zariski writes that "the reader will find in the introduction by M. Artin and B. Mazur an illuminating discussion of these papers and of their impact on later work by other mathematicians. Their discussion includes, in particular, my papers dealing with the following three topics: (1) solvability in radicals of equations of certain plane curves; (2) the fundamental group of the residual space of plane algebraic curves; (3) the topology of the singularities of plane algebraic curves."M. Artin (of MIT) and B. Mazur (of Harvard) both studied under Zariski at Harvard and have since become his colleagues. Their Introduction to this volume provides a useful perspective on Zariski's topological work and on this area of topology in general.These volumes are included in the series Mathematicians of Our Time, under the general editorship of Gian-Carlo Rota. Other volumes in the series now published include papers by Paul Erdos, Einar Hille, Charles Loewner, Percy Alexander MacMahon, George Polya, Hans Rademacher, Stanislaw Ulam, and Norbert Wiener.

Vibrations of Curved Bars Perpendicular to Their Planes

Abstract: We have researched the out-of-plane vibrations of curved bars with uniform cross section, of which center lines are some plane curves Two general methods for solving the problems are presented: one is to use a variable, of which the relation to arc length represents the curvature and the other is to use the curvature expressed by arc length As numerical examples, the frequencies and the mode shapes are shown in graphs for symmetric arc bars with clamped ends having the center lines in the form of ellipses, sines, catenaries, hyperbolas, parabolas and cycloids And from the results, a general tendency in the out-of plane vibrations of curved bars is found

Higher-spin fields in a curved space-time

Abstract: Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.

The isoperimetric theorem for curves on minimal surfaces

Abstract: A short proof is given for a sharpened form of the isoperimetric inequality for curves on minimal surfaces. By following a line of development used by Sachs [7] in treating inequalities for plane curves, one can give an economical formulation to the proof of the isoperimetric theorem for curves on minimal surfaces. Let C be a smooth simple closed curve in Euclidean n-space, where n > 2. In what follows we shall assume, as can be achieved by a translation, that the centroid of arc length of C is at the origin. Hence, if x is the position vector, we assume

Connectedness in topological Hjelmslev planes

Abstract: It is shown that a topological affine Hjelmslev plane is connected or the quasi-component of each point is contained in its neighbour class. If one neighbour class of a point is connected, then they all are, and each is equal to the quasi-component and the component of the point. For topological projective Hjelmslev planes a weaker form of connectedness (∼-connectedness) is defined and it is proved that the plane is ∼-connected or each neighbour class is equal to it ∼-quasi-component. In addition it is shown that the ∼-connectedness of the plane is equivalent to the ∼-connectedness of a line, or other special subsets of the plane, or the connectedness of a line in the associated ordinary plane. Finally it is shown, if the plane is uniform, that ∼-connectedness and connectedness are equivalent and so the plane is either connected, totally disconnected or each neighbour class is equal to the corresponding quasi-component.

Diffraction by a curved half space

Abstract: A method is proposed for finding the asymptotic expansion of the solution and current in a neighborhood of the edge of a curved half plane. For the two-dimensional Dirichlet problem (the incident radiation is assumed to be a plane wave) the first two terms of the expansion are found.

Computable isomorphism invariants for the fundamental group of the complement of a plane projective curve

Abstract: The aim of this paper is to attach computable isomorphism invariants to the fundamental groups r,(P' c) where c is an irreducible plane projective curve. We use these invariants to distinguish certain of these groups. The vehicle used to obtain these invariants is the free differential calculus of R. Fox.