Special Points on the Product of Two Modular Curves

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C², containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1such that X is the image, under the usual map, of the modular curve Y₂0(m). The proof uses some number theory and some topological arguments.     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

Analysis of the Xedni Calculus Attack

The xedni calculus attack on the elliptic curve discrete logarithm problem (ECDLP) involves lifting points from the finite field to the rational numbers and then constructing an elliptic curve over that passes through them. If the lifted points are linearly dependent, then the ECDLP is solved. Our purpose is to analyze the practicality of this algorithm. We find that asymptotically the algorithm is virtually certain to fail, because of an absolute bound on the size of the coefficients of a relation satisfied by the lifted points. Moreover, even for smaller values of p experiments show that the odds against finding a suitable lifting are prohibitively high.     

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

We consider the quotient of the Hermitian curve defined by the equation y^q + y = x^m over where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of -rational points on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m=q +1 in the above equation Communicated by: J.W.P. Hirschfeld     

On the spectra of finite-dimensional perturbations of matrix multiplication operators

Let be a closed rectifiable curve and a region in the complex plane. Suppose for each , R() represents multiplication by an nxn-matrix of rational functions and F() is a finite rank operator, both acting on the Hilbert space L ₂ ⁿ (). Sufficient conditions are given for the integer valued function dim ker (R()+F()) to be continuous at all but finitely many points in . This result is applied to singular integral operators.This work was partially supported by the National Science Foundation.     

Die stabile Reduktion der Fermatkurve über einem Zahlkörper

Let p be an odd prime and F the Fermat curve of degree p, defined by x^p+y^p=1 over . Although the curve F has bad reduction at the prime (p), the stable reduction theorem assures that over some number field K/ we can get stable reduction of the curve F at the primes lying above p. We have determined it in this paper. See Abb.1.     

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s 2n points in general linear position in P ⁿ , and the Koszul property of the generic Artinian Gorenstein algebra of socle degree 3.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Large deflections of eccentrically loaded arches

Zusammenfassung Auf Grund der Hypothesen von Ebenbleiben und Normalität der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogenträger abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogenträgers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nähe der kritischen Last bewirken kleine Exzentrizitäten bedeutende Grössenänderungen der Spannungsresultierenden und der Durchbiegungen.

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Notation A cross-sectional area of curved beam - a radius of centroidal circle - E modulus of elasticity - e eccentricity of the load (Fig. 2) - F an arbitrary function - H horizontal component of the internal forceR acting on a cross section of the arch rib (Fig. 2) - h _P horizontal displacement of the loadP (Fig. 2) - I moment of inertia of the cross-sectional area - k ² =4p ²/(1+4p ² sin²₀) - L span (distance between supports),L=2a sin - M internal bending couple (Figs. 1 and 2) - N internal normal tensile force (Figs. 1 and 2) - n distributed tangential load (Fig. 1) - P downward point load (Fig. 2) - p ² –R a ² /E I - Q internal shearing force (Figs. 1 and 2) - q distributed normal load (Fig. 1) - R internal resultant force (Fig. 2);R ²=H ²+V ²=N ²+Q ² - radius of curvature of the undeformed centroidal curve - s length along the unextended centroidal curve measured from the left support - length along the unextended centroidal curve measured from the right support - u tangential displacement component of the centroidal curve (Fig. 1) - V vertical component ofR (Fig. 2) - v _P vertical displacement of the loadP (Fig. 2) - w normal displacement component (Fig. 1) - x, y rectangular coordinates of the deformed left portion of the centroidal curve (Fig. 2) - Z - z normal distance (positive inward) from centroidal curve (Fig. 1) - half subtending angle of the arch (Fig. 2) - angle of rotation of the centroidal curve (Fig. 1) - extensional strain of the centroidal curve - ^z extensional strain of the linez=constant - y cos–x sin - angle between the tangent to the formed left portion of the centroidal curve and the horizontal (Fig. 2) - (u–w)/r, whereu=du/dø - angle betweenH andR - x cos+y sin - normal stress along the centroidal curve - ^z normal stress along the linez=constant - angle measured from the radius at the left support of the undeformed arch - (–)/2 (Fig. 2) - (+u)/r, where =d/dø A bar over a letter indicates that the entity pertains to the right portion of the arch. Asterisk indicates the deformed configuration. Primes indicate derivatives with respect to ø.     

On complex projective surfaces with trigonal hyperplane sections

Let S be a complex projective surface endowed with an ample and spanned line bundle L. Assume that (S,L) does not belong to some special classes and that c_l(L)²10. We prove that(K_SL)·K_S–3 and |L| contains a trigonal curve (of genus4) iff either (S,L) is a rational surface ruled by cubics, or the g¹ ₃ of C is cut out by |K_S ^–1|. This result applies to surface having a hyperplane section which is a trigonal curve.Partially supported by the M.P.I. of the Italian Government     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Über Zindler-Kurven in euklidischen Räumen

Zindler-curves in Euclidean plane ² are closed curves with a one-parametric set of congruent main chords which bisect both the length and the enclosed area of the curve. Related curves z ⁿ have been studied by J.Hoschek [2], [3] (in the case n=3) and B.Wegner [8] (for arbitrary n2). In this note we generalize the results on Zindler-curves in ⁿ. These curves can simply be generated since the midpoints of the main chords are situated on the striction curve of the main chord-surface.     

Estimates for lebesgue constants in dimension two

Summary Let D denote the interior of a piecewise regular curve of R² having a point with Gauss curvature different from zero. We show that the Lebesgue constants L ^D relative to D behave like ^1/2 as .     

Une Dualité Pour Les Courbes Hyperconvexes

An hyperconvex curve is a curve ξ¹ in such that any n distinct points of the curve are in direct sum. We give here a property of duality of those curves when they admit furthermore an osculating flag. Namely if is a continuous curve of osculating flags of an hyperconvex curve ξ¹, we prove that the curve ξ^n-1 is hyperconvex too and admit a curve of osculating flags.Mathematics Subject Classiffications (2000). 22E40, 53C35     

A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a construction and various possible extensions of the method.      

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Some characterizing properties of the simplex

We shall prove that a convex body in ^d (d2) is a simplex if, and only if, each of its Steiner symmetrals is a convex double cone over the symmetrization space or, equivalently, has exactly two extreme points outside of this hyperplane. In [3] it is shown that every Steiner symmetral of an arbitrary d-simplex is such a double cone, more precisely a bipyramid. Therefore our main aim is to prove that a convex body which is not a simplex has Steiner symmetrals with more than two extreme points not in the symmetrization space. Some equivalent properties of simplices will also be given.     

Maximum of the fourth diameter in the family of continua with prescribed capacity

We obtain a complete solution of the problem of the maximum of the fourth diameter in the family of continua with capacity 1. Let E(o, eⁱ, e^–i). 0<i, e^–i; H(=cap E(o, eⁱ, e^–i). Let C() be the common point of three analytic arcs which form E(o, eⁱ, e^–i). One shows that the indicated maximum is realized by the continuum ={z:H(₀)z ²E(o, eⁱ, e^–i)} where ₀, o<₀z eⁱ z+C ( is a real and C is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 60–79, 1976.     

The geometry of surfaces in 4-space from a contact viewpoint

We study the geometry of the surfaces embedded in ⁴ through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ⁴ has inflection points.The research of the second author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.The research of the third author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.