Showing papers on "Divisor published in 2008"

Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits

Abstract: We study the problem of finding solutions to linear equations modulo an unknown divisor p of a known composite integer N An important application of this problem is factorization of N with given bits of p It is well-known that this problem is polynomial-time solvable if at most half of the bits of p are unknown and if the unknown bits are located in one consecutive block We introduce an heuristic algorithm that extends factoring with known bits to an arbitrary number n of blocks Surprisingly, we are able to show that ln (2) ≈ 70% of the bits are sufficient for any n in order to find the factorization The algorithm's running time is however exponential in the parameter n Thus, our algorithm is polynomial time only for $n = {\mathcal O}(\log\log N)$ blocks

Problems on Minkowski sums of convex lattice polytopes

Abstract: This paper was submitted to the Oberwolfach Conference "Combinatorial Convexity and Algebraic Geometry", October 1997. Let $M={\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\mathbb R}^r$, when is $(M \cap P)+ (M \cap P') = M \cap (P + P')$? Without any additional condition, the equality obviously does not hold. When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity of the canonical map \[ H^0(X,{\mathcal O}_X(D))\otimes H^0(X,{\mathcal O}_X(D'))\to H^0(X,{\mathcal O}_X(D+D')).\] When $X$ is nonsingular, the map is hoped to be surjective, but this remains to be an open question after more than ten years. The paper explores various variations on the question in terms of toric geometry.

On generalized modular forms and their applications

Abstract: We study the Fourier coefficients of generalized modular forms $f(\tau)$ of integral weight $k$ on subgroups $\Gamma$ of finite index in the modular group. We establish two Theorems asserting that $f(\tau)$ is constant if $k = 0$, $f(\tau)$ has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that $f(\tau)$ has a cuspidal divisor, $k$ is arbitrary, and $\Gamma = \Gamma_{0}(N)$, where we show that $f(\tau)$ is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

Coset bounds for algebraic geometric codes

Abstract: For a given curve X and divisor class C, we give lower bounds on the degree of a divisor A such that A and A-C belong to specified semigroups of divisors. For suitable choices of the semigroups we obtain (1) lower bounds for the size of a party A that can recover the secret in an algebraic geometric linear secret sharing scheme with adversary threshold C, and (2) lower bounds for the support A of a codeword in a geometric Goppa code with designed minimum support C. Our bounds include and improve both the order bound and the floor bound. The bounds are illustrated for two-point codes on general Hermitian and Suzuki curves.

Intersections on tropical moduli spaces

Abstract: This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\mathbb{P}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).

On higher-power moments of $\Delta(x)$(II)

Abstract: Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^T\Delta^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for some other well-known error terms in the analytic number theory.

Pointed trees of projective spaces

Abstract: We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of $T_{d,i}$ for $i < n$, it has an inductive construction analogous to but differing from Keel's for $\bar{M}_{0,n}$ which can be used to describe its Chow groups, Chow motive and Poincar\'e polynomials, generalizing \cite{Keel,Man:GF}. We give a presentation of the Chow rings of $T_{d,n}$, exhibit explicit dual bases for the dimension 1 and codimension 1 cycles. The variety $T_{d,n}$ is embedded in the Fulton-MacPherson spaces $X[n]$ for \textit{any} smooth variety $X$ and we use this connection in a number of ways. For example, to give a family of ample divisors on $T_{d,n}$ and to give an inductive presentation of the Chow groups and the Chow motive of $X[n]$ analogous to Keel's presentation for $\bar{M}_{0,n}$, solving a problem posed by Fulton and MacPherson.

Long-reach ethernet for 1000base-t and 10gbase-t

Abstract: A physical-layer device (PHY) having corresponding methods comprises: a data rate module to select a data rate divisor N, where N is at least one of a positive integer, or a real number greater than, or equal to, 1; and a PHY core comprising a PHY transmit module to transmit first signals a data rate of M/N Gbps, and a PHY receive module to receive second signals at the data rate of M/N Gbps; wherein the first and second signals conform to at least one of 1000BASE-T, wherein M = 1, and 10GBASE-T, wherein M = 10.

Logarithmic vector fields along smooth divisors in projective spaces

Abstract: We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type.

Nef divisors on $\bar{M}_{0,n}$ from GIT

Abstract: We introduce and study the GIT CONE of $\bar{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n//SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\bar{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson arXiv:0709.4037. (Cf. also a different proof by Fedorchuk and Smyth arXiv:0810.1677)

Efficient Algorithms for Integer Division by Constants Using Multiplication

Abstract: We present a complete analysis of the integer division of a single unsigned dividend word by a single unsigned divisor word based on double-word multiplication of the dividend by an inverse of the divisor. The well-known advantage of this method yields run-time efficiency, if the inverse of the divisor can be calculated at compile time, since multiplication is much faster than division in arithmetic units. Our analysis leads to the discovery of a limit to the straightforward application of this method in the form of a critical dividend, which fortunately associates with a minority of the possible divisors (20%) and defines only a small upper part of the available dividend space. We present two algorithms for ascertaining whether a critical dividend exists and, if so, its value along with a circumvention of this limit. For completeness, we include an algorithm for integer division of a unsigned double-word dividend by an unsigned single-word divisor in which the quotient is not limited to a single word and the remainder is an intrinsic part of the result.

Γ-reduction for smooth orbifolds

Abstract: The aim of this short note is to show how to construct a rational Remmert reduction (the \({\widetilde \Gamma}\) -reduction) for the universal cover of smooth orbifolds (X/Δ). Doing this, we are led to introduce some singular Kahler metrics on (X/Δ) adapted to the \({\mathbb {Q}}\) -divisor Δ.

Algebraic Stein Varieties

Abstract: It is well-known that the associated analytic space of an affine variety defined over $\mathbb{C}$ is Stein but the converse is not true, that is, an algebraic Stein variety is not necessarily affine. In this paper, we give sufficient and necessary conditions for an algebraic Stein variety to be affine. One of our results is that an irreducible quasi-projective variety $Y$ defined over $\mathbb{C}$ with dimension $d$ ($d\geq 1$) is affine if and only if $Y$ is Stein, $H^i(Y, {\mathcal{O}}_Y )=0$ for all $i>0$ and $\kappa(D, X)= d$ (i.e., $D$ is a big divisor), where $X$ is a projective variety containing $Y$ and $D$ is an effective divisor with support $X-Y$. If $Y$ is algebraic Stein but not affine, we also discuss the possible transcendental degree of the nonconstant regular functions on $Y$. We prove that $Y$ cannot have $d-1$ algebraically independent nonconstant regular functions. The interesting phenomenon is that the transcendental degree can be even if the dimension of $Y$ is even and the degree can be odd if the dimension of $Y$ is odd.

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Abstract: Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=\lambda \] where $\lambda>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $\lambda$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$.

An Isodiametric Problem for Equilateral Polygons

Abstract: . The maximal perimeter of an equilateral convex polygon with unitdiameter and n = 2 m edges is not known when m ≥ 4. Using experimentalmethods, we construct improved polygons for m ≥ 4, and prove that theperimeters we obtain cannot be improved for large n by more than c/n 4 , fora particular constant c. 1. IntroductionThere are two classical isodiametric problems for curves in the plane. First,determine the maximal area enclosed by a closed planar curve with unit diameter.Second, determine the maximal perimeter of a closed, convex planar curve with unitdiameter. It is well known that the unique solution in both problems is attained bya circle. The area problem was solved by Bieberbach in 1915 [8], and the perimeterquestion was answered by Rosenthal and Sz´asz in 1916 [17].We can ask the same questions for polygons with a fixed number of sides.First, determine the maximal area A(P) enclosed by a polygon P with n sidesand unit diameter. Second, determine the maximal perimeter L(P) of a convexpolygon P with n sides and unit diameter. These problems were first investigatedby Reinhardt in 1922 [16]. In that article, Reinhardt established several results onthese problems, which we briefly summarize here.• If n is odd, then the regular n-gon with unit diameter attains both themaximal area and the maximal perimeter.• The regular n-gon is the unique solution in the area problem when n isodd, but it is the unique solution in the perimeter problem when n is oddonly if n is prime.• If n is even and n ≥ 6, then the regular n-gon with unit diameter doesnot attain the maximal area, nor the maximal perimeter.• If n has an odd divisor, then every polygon with maximal perimeter isequilateral.Proofs of these statements, along with some additional history and background onthese problems, may be found in [14].

Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for $\bar{M}_{0,n}$

Abstract: We introduce a new technique for proving positivity of certain divisor classes on $\bar{M}_{0,n}$ and its weighted variants. Our methods give an unconditional description of the spaces of symmetric weighted pointed rational curves as log canonical models of $\bar{M}_{0,n}$.

HECC Goes Embedded: An Area-efficient Implementation of HECC

Abstract: In this paper we describe a high performance, area-efficient implementation of Hyperelliptic Curve Cryptosystems over GF(2 m ). A compact Arithmetic Logic Unit (ALU) is proposed to perform multiplication and inversion. With this ALU, we show that divisor multiplication using affine coordinates can be efficiently supported. Besides, the required throughput of memory or Register File (RF) is reduced so that area of memory/RF is reduced. We choose hyperelliptic curves using the parameters h(x) = x and \(f(x)=x^5+f_3x^3+x^2+f_0\). The performance of this coprocessor is substantially better than all previously reported FPGA-based implementations. The coprocessor for HECC over GF(283) uses 2316 slices and 2016 bits of Block RAM on Xilinx Virtex-II FPGA, and finishes one scalar multiplication in 311 μs.

On the Euler function of repdigits

Abstract: For a positive integer n we write φ(n) for the Euler function of n. In this note, we show that if b > 1 is a fixed positive integer, then the equation $$ \varphi \left( {x\frac{{b^n - 1}} {{b - 1}}} \right) = y\frac{{b^m - 1}} {{b - 1}}, where x,y \in \{ 1, \ldots ,b - 1\} , $$ has only finitely many positive integer solutions (x, y, m, n).

Donaldson-Thomas theory of A_n x P^1

Abstract: We study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra \hat{gl}(n+1) on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n.

A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period

Abstract: We introduce a class of sparse matrices U m (A p 1 ) of order m by m, where m is a composite natural number, p 1 is a divisor of m, and A p 1 is a set of nonzero real numbers of length p 1. The construction of U m (A p 1 ) is achieved by iteration, involving repetitive dilation operations and block-matrix operations. We prove that the matrices U m (A p 1 ) are invertible and we compute the inverse matrix (U m (A p 1 ))−1 explicitly. We prove that each row of the inverse matrix (U m (A p 1 ))−1 has only two nonzero entries with alternative signs, located at specific positions, related to the divisors of m. We use the structural properties of the matrix (U m (A p 1 ))−1 in order to build a nonlinear estimator for prediction of nearly periodic time series of length m with fixed period.

On solutions of the Schlesinger equation in the neigborhood of the Malgrange Θ-divisor

Abstract: We say that the family (1) is isomonodromic if, for all a ∈ D(a0), the monodromies χa : π1(C \ {a1, . . . , an}) → G = GL(p,C) of the corresponding system are equal to each other. (Under small variations of the parameter a, there exists a canonical isomorphism of the fundamental groups π1(C \ {a1, . . . , an}) and π1(C \ {a1, . . . , an}) generating the canonical isomorphism Hom(π1(C \ {a1, . . . , an}), G)/G ∼= Hom(π1(C \ {a1, . . . , an}), G)/G of the spaces of classes of the duality representations for these fundamental groups; this allows one to compare χa for various a ∈ D(a0).) For example, if the matrix Bi(a) satisfies the Schlesinger equation dBi(a) = − n ∑

On a variant of the large sieve

Abstract: We introduce a variant of the large sieve and give an example of its use in a sieving problem Take the interval [N] = {1,,N} and, for each odd prime p <= N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some interval I_p of Z/pZ of length (p-1)/2 Let A be the set that remains: then |A| << N^{1/3 + o(1)}, a bound which improves slightly on the bound of |A| << N^{1/2} which results from applying the large sieve in its usual form This is a very, very weak result in the direction of a question of Helfgott and Venkatesh, who suggested that nothing like equality can occur in applications of the large sieve unless the unsieved set is essentially the set of values of a polynomial (eg A is the set of squares) Assuming the ``exponent pairs conjecture'' (which is deep, as it implies a host of classical questions including the Lindel\"of hypothesis, Gauss circle problem and Dirichlet divisor problem) the bound can be improved to |A| << N^{o(1)} This raises the worry that even reasonably simple sieve problems are connected to issues of which we have little understanding at the present time

A novel VLSI iterative divider architecture for fast quotient generation

Abstract: In this paper, a novel VLSI iterative divider architecture for fast quotient generation that is based on radix-2 non-restoring division is proposed. To speed up the quotient generation, our method makes use of the magnitude difference between the partial dividend and the divisor for the next iteration so that the proper weight of the quotient can be obtained more rapidly than the conventional methods. Our proposed architecture is very simple compared to the multiplication-based methods such as those that are based on Newton-Raphson. Simulation results show that our proposed method can achieve less than half the number of iterations required by the conventional division (i.e. less than nil vs. n, where n is the bit-width of the dividend and the divisor). The proposed architecture has been synthesized in 0.13 mum CMOS standard cell library to demonstrate the delay and the power efficiency.

Consecutive integers with equally many principal divisors

Abstract: Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa \ n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \ can be read as "/?fl is a principal divisor of n." The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and