Showing papers on "Divisor published in 2002"

Arithmetic on superelliptic curves

Abstract: This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn = c(x) which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.

Stable maps and branch divisors

Abstract: We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.

Higher correlations of divisor sums related to primes III: k-correlations

Abstract: We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.

Fast Genus Three Hyperelliptic Curve Cryptosystems

Abstract: This paper presents a fast addition algorithm for the divisor class groups of genus three hyperelliptic curves. This algorithm improves the most recently proposed Harley algorithm for genus three hyperelliptic curves, which have brought up a noticeable progress since the well known Cantor algorithm. In this paper, we extend the Harley algorithm to genus three curves. The computational cost of the proposed algorithm is I +8 1M for an addition and I +7 4M for a doubling. (Here I and M denote the cost of an inversion and a multiplication on the definition fields.) We also show the implementation of the algorithm on Alpha 21264 / 667MHz, which takes 932 µs for a 160bit scalar multiplication on a divisor class group.

The Module Df s for Locally Quasi-homogeneous Free Divisors

Abstract: We find explicit free resolutions for the D-modules Dfs and D[s] fs/ D[s] fs+1, where f is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper.

Logarithmic cohomology of the complement of a plane curve

Abstract: Let D, x be a plane curve germ. We prove that the complex \( \Omega^\bullet(\log D)_x \) computes the cohomology of the complement of D, x only if D is quasihomogeneous. This is a partial converse to a theorem of [5], which asserts that this complex does compute the cohomology of the complement, whenever D is a locally weighted homogeneous free divisor (and so in particular when D is a quasihomogeneous plane curve germ). We also give an example of a free divisor \( D\subset \mathbb {C}^3 \) which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.

Kawamata-Viehweg vanishing as Kodaira vanishing for stacks

Abstract: We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as Kodaira vanishing for stacks.

Digital frequency divider

Abstract: A system and method is presented for dividing a reference clock frequency by any real number. The invention allows for a real number divisor that could have any desired degree of precision. Additionally, the invention seeks to minimize hardware complexity in realizing such a reference-clock frequency divider. In one particular embodiment of the invention, a system and method is presented, wherein the real number divisor is a real number having a repeating decimal (i.e., the real number may be represented by a fraction).

Higher dimensional Zariski decompositions

Abstract: Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of $\alpha$. The negative part of $\alpha$ is then defined as the real effective divisor $N(\alpha)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $\alpha$ along $D$, and we get in that way a divisorial Zariski decomposition of $\alpha$ into the sum of a class $Z(\alpha)$ which is nef in codimension 1 and the class of its negative part $N(\alpha)$, which is exceptional in the sense that it is very rigidly embedded in $X$. The positive parts $Z(\alpha)$ generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kahler manifold in some detail: under the intersection form (resp. the Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors; our divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kahler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series $|kL|$ as $k\to\infty$.

Method and apparatus for executing division

Abstract: The invention relates to a method and an electronic apparatus implementing the method of executing division. In the method, an auxiliary divisor is retrieved from a look-up table stored in the electronic apparatus, the auxiliary divisor being a predetermined number generated by the product of the powers of the integer two and the reciprocal of the divisor. In the method, the division is executed in the electronic apparatus by multiplying the dividend of the division by the auxiliary divisor. The result of the division is scaled in the electronic apparatus in order to represent it in the desired form by shifting the result obtained by multiplying.

Blow-ups of the Toda lattices and their intersections with the Bruhat cells

Abstract: We study the topology of the set of singular points (blow-ups) in the solution of the nonperiodic Toda lattice defined on real split semisimple Lie algebra $\mathfrak g$. The set of blow-ups is called the Painlev\'e divisor. The isospectral manifold of the Toda lattice is compactified through the companion embedding which maps themanifold to the flag manifold associated with the underlying Lie algebra $\mathfrak g$. The Painlev\'e divisor is then given by the intersections of the compactified manifold with the Bruhat cells in the flag manifold. In this paper, we give explicit description of the topology of the Painlev\'e divisor for the cases of all the rank two Lie algebra, $A_2,B_2, C_2, G_2$, and $A_3$ type. The results are obtained by using the Mumford system and the limit matrices introduced originally for the periodic Toda lattice. We also give a Lie theoretic description of the Painlev\'e divisor of codimension one case, and propose several conjecturesfor the general case.

High-speed programmable frequency-divider with synchronous reload

Abstract: A programmable-divider provides a lower-speed transition signal to effect a synchronized load of a new divisor value during a safe-load period of the programmable-divider, such that the division occurs using either the prior divisor value or the new divisor value, only. A combination of in-phase 120 and reverse-phase 230 counter stages are used to position the divisor-independent period of each counter stage so that an edge of at least one of the lower-speed counter-enabling signals occurs during a period when all of the counter stages are in a divisor-independent period. The preferred selection of in-phase and reverse-phase counter stages also maximizes the critical path duration, to allow for the accurate division of very high speed input frequencies.

Factorization properties of Krull monoids with infinite class group

Abstract: For a non-unit a of an atomic monoid H we call LH(a) = {k ∈ N | a = u1 . . . uk with irreducible ui ∈ H} the set of lengths of a. Let H be a Krull monoid with infinite divisor class group such that each divisor class is the sum of a bounded number of prime divisor classes of H. We investigate factorization properties of H and show that H has sets of lengths containing large gaps. Finally we apply this result to finitely generated algebras over perfect fields with infinite divisor class group.

Restricted divisor sums

Abstract: where the function f(n) is n, n or n + n + p+1 4 , where p ≡ 3 mod 4 is a rational prime, and where dα(n) = #{d : d|n and 1 ≤ d ≤ α} for real α ≥ 1. Motivation for considering these sums comes from an expression which is derived for the class number of a quadratic field with discriminant −p, in terms of a certain restricted divisor sum. This sum is currently too difficult to estimate, in that the restrictions on divisors depend on the summation variable n. In deriving asymptotic expressions for the sum ∑

Some open problems related to small divisors

Abstract: 0 Introduction 1 One-Dimensional Small Divisor Problems (On Holomorphic Germs and Circle Diffeomorphisms) 1.1 Linearization of the quadratic polynomial. Size of Siegel disks 1.2 Herman rings. Differentiable conjugacy of diffeomorphisms of the circle 1.3 Gevrey classes 2 Finite-Dimensional Small Divisor Problems 2.1 Linearization of germs of holomorphic diffeomorphisms of \((\mathbb{C}^n, 0)\) 2.2 Elliptic fixed points and KAM theory 2.3 \(\mathbb{Z}^k\)-actions 2.4 Diffeomorphisms of compact manifolds 3 KAM Theory and Hamiltonian Systems 3.1 Twist maps 3.2 Euler-Lagrange flows 3.3 n-body problem 4 Linear Quasiperiodic Skew-Products, Spectral Theory and Hamiltonian Partial Differential Equations 4.1 Reducibility of skew-products 4.2 Spectral theory and integrated density of states 4.3 Nonlinear Hamiltonian PDEs References

Methods and apparatus for binary division using look-up table

Abstract: Improved methods of operating a digital data processor to perform binary division include estimating reciprocals of at least selected divisors based on value accessed from a look-up table. For divisors in a first numerical range, the estimation can be based on a value stored in a first look-up table at an index defined by the divisor. For divisors in a second numerical range, the estimation can be based on an index that is a bitwise-shifted function of the divisor. The methods can be applied to scalar and vector binary division.

Pseudo random signal producing circuit

Abstract: A pseudo random signal producing circuit includes a generator 110 for generating a first pseudo random signal having a bit width a (a being an integer not smaller than 1), a generator 120 for generating a second pseudo random signal having a bit width b (b being an integer not smaller than 1 and different from a), a matrix calculator 130 for carrying out matrix calculation upon the first and the second pseudo random signals to produce a calculation result signal having a bit width (a*b), an N-bit shift register 200 responsive to the calculation result signal having the bit width (a*b) for producing an output pseudo random signal having a bit width N (N being a divisor of (a*b)), and a frequency-division clock generator 300 for driving a pseudo random data generator 100.

Glitch-free frequency dividing circuit

Abstract: The present invention discloses a glitch-free frequency dividing circuit, comprising: a frequency dividing module, dividing the frequency of a reference pulse according to the divisor, outputting a frequency divided output pulse and receiving a control signal such that the state of the frequency divided output pulse is maintained the same when the control signal is enabled; and a latch module, detecting the state of the frequency divided output pulse after a divisor switching signal is received, enabling the control signal when the frequency divided output pulse is as pre-determined, switching the divisor when the frequency divided output pulse is as pre-determined and disabling the control signal after the divisor is switched; whereby the generation of the glitch is prevented during the switching of the divisor.

Topology of the real part of hyperelliptic Jacobian associated with the periodic Toda lattice

Abstract: This paper concerns the topology of the isospectral {\it real} manifold of the ${\mathfrak sl}(N)$ periodic Toda lattice consisting of $2^{N-1}$ different systems. The solutions of those systems contain blow-ups, and the set of those singular points defines a devisor of the manifold. Then adding the divisor, the manifold is compactified as the real part of the $(N-1)$-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus $g=N-1$. We also study the real structure of the divisor, and then provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based upon the sign-representation of the Weyl group of ${\mathfrak sl}(N)$.

Effective divisor classes on a ruled surface

Abstract: The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron-Severi group. Despite the finite generation of the whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, and the methods used to explicitly calculate this monoid seem to vary widely as one proceeds from one type of surface to another in the standard classification scheme (see Rosoff, 1980, 1981). In this paper we shall use concrete vector bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled surface over a given base curve. We will find that, over a base curve of genus 0, the monoid of effective divisor classes is very simple, having two generators (which is perhaps to be expected), while for a ruled surface over a curve of genus 1, the monoid is more complicated, having either two or three generators. Over a base curve of genus 2 or greater, we will give necessary and sufficient conditions for a ruled surface to have its monoid of effective divisor classes finitely generated; these conditions point to the existence of many ruled surfaces over curves of higher genus for which finite generation fails.

Extended precision integer divide algorithm

Abstract: A method for an extended precision integer divide algorithm. The method of one embodiment comprises separating a first L bits wide integer dividend into two equal width portions, wherein a first integer format portion comprises lower M bits of the first integer dividend and a second integer format portion comprises upper M bits of the first integer dividend, wherein M is equal to ½ L. The first integer format portion is converted into a first floating point format portion. An N bits wide integer divisor is converted from an integer format into a floating point format divisor. The first floating point format portion is divided by the floating point format divisor to obtain a first floating point format quotient. The first floating point format quotient is converted into a first integer format quotient. The second integer format portion is converted into a second floating point format portion. The second floating point format portion is divided by the floating point format divisor to obtain a second floating point format quotient. The second floating point format quotient is converted to a second integer format quotient. The first and second integer format quotients are summed together to generate a third integer format quotient.

A TORELLI THEOREM FOR SPECIAL DIVISOR VARIETIES X ASSOCIATED TO DOUBLY COVERED CURVES ${\tilde C}/C$

Abstract: Let Ci, i=1,2, be two smooth non-hyperelliptic curves over the complex numbers of genus g≥ 3, and $\pi_i: {\tilde C}_i\rightarrow C_i$ connected etale double covers, such that the theta divisors Ξi of the associated Prym varieties (pi, Ξi) are non singular in codimension ≤ 3. If $Nm_i: {\rm Pic}^{2g-2} ({\tilde C}_i)\rightarrow {\rm Pic}^{2g-2} (C_i)$ are the norm maps, then Ξi is isomorphic to {$L\ \mbox{in\ Pic}^{2g-2} ({\tilde C}_i): Nm_i (L) = \omega_{C_i}$ and h0 (L) is even and positive}. Then the Abel maps define generic ℙ1 bundles Xi→Ξi, where Xi is the special divisor variety $X_i = \{D\ \mbox{in}\ {\tilde C}_i^{(2g - 2)}: Nm_i ({\mathcal O}(D)) = \omega_{C_i}$ and $h^0({\mathcal O}(D))$ even}. We prove, under the hypotheses above, that biregular isomorphism of the special divisor varieties X1≅ X2 implies isomorphism of the double covers ${\tilde C}_1/C_1\cong {\tilde C}_2/C_2$.

The module $D f^s$ for locally quasi-homogeneous free divisors

Abstract: We find explicit free resolutions for the $\scr D$-modules ${\scr D} f^s$ and ${\scr D}[s] f^s/{\scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper

A new proof of a theorem of Ramanujam-Morrow

Abstract: Morrow [9] classified all weighted dual graphs of the boundary of the minimal normal compactifications of the affine plane $\mathbf{A}^{2}$ by using a result of Ramanujam [10] that any minimal normal compactification of $\mathbf{A}^{2}$ has a linear chain as the graph of the boundary divisor. In this article, we give a new proof of the above-mentioned results of Ramanujam-Morrow [9] from a different point of view and by the purely algebro-geometric arguments. Moreover, we show that the affine plane $\mathbf{A}^{2}$ is characterized by the weighted dual graph of the boundary divisor.

Topology of the Real Part of the Hyperelliptic Jacobian Associated with the Periodic Toda Lattice

Abstract: We study the topology of the isospectral real manifold of the \(sl(N)\) periodic Toda lattice consisting of 2N−1 different systems. The solutions of these systems contain blow-ups, and the set of these singular points defines a divisor of the manifold. With the divisor added, the manifold is compactified as the real part of the (N−1)-dimensional Jacobi variety associated with a hyperelliptic Riemann surface of genus g=N−1. We also study the real structure of the divisor and provide conjectures on the topology of the affine part of the real Jacobian and on the gluing rule over the divisor to compactify the manifold based on the sign representation of the Weyl group of \(sl(N)\).

Stickelberger ideals and divisor class numbers

Abstract: Let K/k be a finite abelian extension of function fields with Galois group G. Using the Stickelberger elements associated to K/k studied by J. Tate, P. Deligne and D. Hayes, we construct an ideal I in the integral group ring $\mathbb{Z}[G]$ relative to the extension K/k, whose elements annihilate the group of divisor classes of degree zero of K and whose rank is equal to the degree of the extension. When K/k is a (wide or narrow) ray class extension, we compute the index of I in $\mathbb{Z}[G]$ , which is equal to the divisor class number of K up to a trivial factor.

On a generalized divisor problem. I

Abstract: We give a discussion on the properties of $\Delta_a(x)$ ($-1

On the divisor products and proper divisor products sequences

Abstract: Let n be a positive integer, pd(n) denotes the product of all positive divisors of n, qd(n) denotes the product of all proper divisors of n. In this paper, we study the properties of the sequences {pd(n)} and {qd(n)}, and prove that the Makowski & Schinzel conjecture hold for the sequences {pd(n)} and {qd(n)}.

On Partition Functions and Divisor Sums

Abstract: Let n;r be natural numbers, with r‚ 2. We present convolution-type formulas for the number of partitions of n that are (1) not divisible by r; (2) coprime to r. Another result obtained is a formula for the sum of the odd divisors of n.

Effective divisors on $M_g$ and a counterexample to the Slope Conjecture

Abstract: We prove two statements on the slopes of effective divisors on the moduli space of stable curves of genus g: first that the Harris-Morrison Slope Conjecture fails for g=10 and second, that in order to compute the slope of the moduli space of curves for g\leq 23, one only has to consider the coefficients of the Hodge class and that of the boundary divisor \delta_0 in the expansion of the relevant divisors. We conjecture that the same statement holds in arbitrary genus.