Showing papers on "Divisor published in 1996"

Approximate polynomial greatest common divisors and nearest singular polynomials

Abstract: The problem of computing the greatest common divisor (gcd) of two polynomials ~, g ~ A[z], A being a unique factorization domain, is well understood and there area number of efficient algorithms for computing polynomial gcds beginning with the the work of Collins and Brown [3, 4, 9]. In this paper, we investigate the problem of finding approximate gcds. Given a pair of polynomials ~, g with real/complex coefficients, we wish to determine a small perturbation of the coefficients of ~, g such that the perturbed polynomials have a non-trivial gtd. We treat several variations of this problem and apply our techniques to the problem of finding a polynomial having multiple roots clo~seto a given polynomial. In this paper, F[a, b] denotes the polynomial ring in a, b over the field ~ and F(a, b) denotes the field of rational functions in a, b over f. C denotes the field of complex numbers and ‘R denotes the field of real numbers. For a polynomial j = f.z” + j.–lz”–l + . . . + fo ~ C[x], 11~11denotes

Complexity of the Frobenius problem

Abstract: Consider the Frobenius Problem: Given positive integersa1,...,an withai ≥ 2 and such that their greatest common divisor is one, find the largest natural number that is not expressible as a non-negative integer combination ofa1,...,an. In this paper we prove that the Frobenius problem is NP-hard, under Turing reductions.

Defects for Ample Divisors of Abelian Varieties, Schwarz Lemma, and Hyperbolic Hypersurfaces of Low Degrees

Abstract: The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let $f:{\bf C}\rightarrow A$ be a holomorphic map. Then the defect for the map $f$ and the divisor $D$ is zero. Corollary to Main Theorem. The complement of an ample divisor $D$ in an abelian variety $A$ is hyperbolic in the sense that there is no nonconstant holomorphic map from $\bf C$ to $A-D$.

Weak Hironaka theorem

Abstract: The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.

Weak Hironaka theorem

Abstract: The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.

Contractible Extremal Rays on \overline{M}_{0,n}

Abstract: We consider the cones of curves and divisors on the moduli space of stable pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n, which is a moduli space of pairs. We find generators for contractible extremal rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n). This second cone turns out to be simplicial. We give complete descriptions of NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the second). We also have results of independent interest on when curves in a divisor generate the cone of curves of the ambient variety.

Some analogues of smith's determinant

Abstract: We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants.

Divisors on Generic Complete Intersections in Projective Space

Abstract: Let V be a generic complete intersection of hypersurfaces of degree dl, d2, , dm in n-dimensional projective space. We study the question when a divisor on V is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case dl + d2 + + dm = n + 2. 0. INTRODUCTION Let V be a generic complete intersection of hypersurfaces of degree dl, d2, , dm in P'. A conjecture of Kobayashi (cf. [L]) states that V is hyperbolic if d = d, + d2 + . ?+ dm > n +2. In general, S. Lang [L] has conjectured that a variety X is hyperbolic if and only if every subvariety of X is of general type. In this paper, we will prove the following Theorem 1. Let V be a complete intersection of m generic hypersurfaces of degree di I d2,... , dm in pn, M c V a reduced and irreducible divisor, pg(M) the geometric genus of the desingularization of M. Assume that 1 2 for all i. Then (1) ps(M)>nif d=dl+d2 + +?dm > n+2, (2) Mis of general type if d= di+d2+ +dm> n+2. In [E1,E2], Ein has shown that M is nonrational if d > n + 2, and is of general type if d > n + 2. Here we are going to give an alternative proof of it. Ein also proved that every subvariety of V of dimension 1 is nonrational if d > 2n m 1+ 1, and is of general type if d > 2n m I + 1. Therefore the improvement we made here is in the case d = n + 2 and I = n m 1. In particular, we conclude that the divisor M can not be an abelian variety. If a variety X is hyperbolic, then every rational map of an abelian variety or Pl into X is constant. On the other hand, Lang [L] conjectured that this condition is also sufficient for X to be hyperbolic. If V is a generic hypersurface in pn, it was first shown by Clemens [CKM] that V contains no rational curves, if deg V > n 1. In [Xl], we study generic surfaces in P3, obtain that every curve C on S has geometric genus g(C) > 1d(d 3) 2 (d =deg S), and the bound is sharp. We also obtain results about divisors on a generic hypersurface in pn. In [X2], we generalize these results to some nongeneric cases. Received by the editors August 5, 1995. 1991 Mathematics Subject Classification. Primary 14J70, 14B07. Partially Supported by NSF grant DMS-9401547. (?)1996 American Mathematical Society 2725 This content downloaded from 157.55.39.243 on Thu, 06 Oct 2016 04:56:22 UTC All use subject to http://about.jstor.org/terms

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

Abstract: For 1/2<σ<1 fixed, letEσ(T) denote the error term in the asymptotic formula for\(\int_0^T {|\zeta (\sigma + it)|^2 dt} \). We obtain some new bounds forEσ(T), and an Ω_-result which is the analogue of the strongest Ω_-result in the classical Dirichlet divisor problem.

Projective structures on a Riemann surface

Abstract: For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line bundle $K_{X\times X}\otimes {\cal O}_{X\times X}(2\Delta)$ on $X\times X$, where $K_{X\times X}$ is the canonical bundle of $X\times X$ and $\Delta$ is the diagonal divisor. We show that $L$ has a canonical trivialisation over the nonreduced divisor $2\Delta$. Our main result is that the space of projective structures on $X$ is canonically identified with the space of all trivialisations of $L$ over $3\Delta$ which restrict to the canonical trivialisation of $L$ over $2\Delta$ mentioned above. We give a direct identification of this definition of a projective structure with a definition of Deligne.We also describe briefly the origin of this work in the study of the so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum field theory.

Division algorithm for floating point or integer numbers

Abstract: A computer-implemented algorithm for dividing numbers involves subtracting the divisor from the divided to generate a first intermediate result, which is then shifted by N-bits to obtain a remainder value. A portion of the remainder and a portion of the divisor are utilized to generate one or more multiples from a look-up table, each of which is multiplied by the divisor to generate corresponding second intermediate results. The second intermediate results are subtracted from the remainder to generate corresponding third intermediate results. The largest multiple which corresponds to a third intermediate result having a smallest positive value is the quotient digit. The third intermediate result that corresponds to the largest multiple is the remainder for the next iteration.

Primitive divisors of Lucas and Lehmer sequences, II

Abstract: Let $\al$ and $\be$ be conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\al$ and $\be$ with $\hgt(\be/\al) \leq 4$, the $n$-th element of these sequences has a primitive divisor for $n > 30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

Computer for performing non-restoring division

Abstract: A computer that performs division in either floating point or integer representation according to a novel algorithm in which a divisor is subtracted from a dividend to generate a first intermediate result. A shifter shifts the intermediate result by N-bits, where N is an integer and 2 N is equal to the radix, to obtain a remainder. A look-up table produces one or more multipliers based upon an upper-bit portion of the remainder and an upper-bit portion of the divisor. The divisor is multiplied by each of the one or more multiples to generate second intermediate results. Each of the secondary intermediate results is then subtracted from the remainder to generate one or more corresponding third intermediate results. A current quotient digit is selected as the largest multiplier which corresponds to the third intermediate result having the smallest possible value (as among all of the third intermediate results).

Numerical Criteria for Very Ampleness of Divisors on Projective Bundles Over an Elliptic Curve

Abstract: Let D be a divisor on a projectivized bundle over an elliptic curve. Numerical conditions for the very ampleness of D are proved. In some cases a complete numerical characterization is found.

A note on Behrend sequences

Abstract: The concept of a Behrend sequence is one of the most fundamental and challenging in the theory of sets of multiples. A sequence A of integers exceeding 1 is called a Behrend sequence if almost all integers n have at least one divisor in A, or, in other words, if its set of multiples M(A) = {ma : m 1, a ∈ A} has natural density 1. This was recently defined formally by Hall (1990), but the idea has been constantly used by Erdős in the last half-century. Recent progress on this topic may be found in Hall-Tenenbaum (1992), Erdős-Hall-Tenenbaum (1994), Tenenbaum (1994). By the Davenport-Erdős theorem (1937, 1951), any set of multiples M(A) has a logarithmic density δM(A), equal to its lower asymptotic density, moreover

Division by a constant

Abstract: A ripple through divider of a dividend by a constant is obtained by cascading a plurality of partial quotient tables. Each table incorporates the same divisor, so the divisor need not appear as an input. In one binary integer implementation for an n bit dividend that dividend is represented as n+1 bits having an MSB of 0. If the binary divisor is of k bits, then the most significant k+1 bits are applied to an input of a first partial quotient table. It produces one bit of fractionary quotient that becomes the MSB of the final quotient, and k bits of fractionary remainder. That fractionary remainder is combined as MSB's with an LSB that is the next and most significant unused dividend bit. This forms k+1 inputs to a second partial quotient table. It in turn produces a partial quotient bit that becomes the second most significant final quotient bit, and k-many more fractionary remainder bits. The cascading continues with additional stages of partial quotient tables until all dividend bits have been used. At that level the final quotient is available and the last partial remainder bits are indeed the actual final remainder bits. The partial quotient tables may be look-up tables implemented as ROM's or they may be constructed of discrete gating.

Analog arithmetic circuit

Abstract: An analog arithmetic circuit directly divides an input voltage by another input voltage with high accuracy without requiring a logarithmic conversion process or an adjustment process. The analog arithmetic circuit includes: an integrator for integrating a dividend signal and a feedback signal; a hysteresis comparator having two threshold levels to compare an output signal of the integrator and generates a comparison output; a limiter which receives the comparison output and a divisor signal and generates the feedback signal that is proportional to the divisor signal; an average circuit connected to an output of the hysteresis comparator to generates an average value of the comparison output as a quotient signal.

Problems Similar to the Additive Divisor Problem

Abstract: For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(pr)≤Ar,A>0, and for anye>0 there exist constants\(A_\varepsilon\),α>0 such that\(f(n) \leqslant A_\varepsilon n^\varepsilon\) and Σp≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved: $$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$$ . Hereτ(n) is the number of divisors ofn andC(ƒ) is a constant.

Ciphering device, deciphering device, ciphering and deciphering device and cipher system

Abstract: PURPOSE: To provide a ciphering device and a cipher system particularly excellent in the deciphering speed as compared with RSA ciphers in use. CONSTITUTION: This device is provided with a key generation means 14 which generates primes p and q and at the time of computation with dp and dq satisfying dp =(1/e)mod(p-1), dq =(1/e)mod(q-1), where an integer e is mutually prime with the least common multiple of the product n=pq, (p-1) and (q-1), sets the product n and an integer e to be public keys and sets the primes p, q and dp , dq to be secret keys. In addition the device is provided with a ciphering calculation means which makes an integer pair of inputted plain texts correspond to a point on a cubic curve, determines a point obtained by e-folding the point by the use of the public keys by arithmetic on the cubic curve, and outputs arithmetic results as a cipher text, and a deciphering arithmetic means which subjects the integer pair of the inputted cipher text to homomorphic transformation, then raises the result to the dp -th power under a divisor p and dq -th power under a divisor q, and synthesizes them by the use of the Chinese remainder theorem to output a plain text.

On the existence of good divisors on Fano varieties of coindex 3

Abstract: We prove that the linear system $|-1/3K_X| on a non-singular Fano fivefold $X$ of index 3 contains an irreducible divisor with only canonical singularities.

Interpolation circuit for measuring tool location w.r.t. workpiece

Abstract: The equipment includes a waveform unit (6) to form the modulated wave from the first unit (4), an addition unit, and the low-pass filter (5). The result goes to a bandpass filter (7) to device the n-th higher harmonic signal components and these pass to a mixers (8) and (11) so that these harmonics can be combined with a signal of a frequency n times higher than the modulated wave and zero phase or with a similar signal of 90 degree phase. There are then two low-pass filters (9) and (12) and finally a voltage comparator and interpolator (13) taking the sine n theta and the cos n theta signals from the filters and interpolating A phase and B phase signals, using a divisor equal to n x m.

Method and apparatus for computing a real time clock divisor

Abstract: A method and apparatus for generating a divisor having a variable value for real time calculations. A microprocessor's time base register is used to generate the divisor. All interrupts and refreshes are disabled on the microprocessor before calculating the divisor. The microprocessor's time base register is cleared and allowed to count the number of clock ticks in a one second interval. The time base register is read and the value used as a divisor for real time calculations.

Recording method and apparatus for control track

Abstract: A method and apparatus for recording control pulses on a control track are provided. The control pulses correspond to a number M (where M is an integer equal to or larger than four) of azimuthal tracks on the tape on which the number M of segmented data, obtained by dividing image data corresponding to one frame into the number M of segments, are recorded. One period of the control pulses corresponds to a number N (where N is a divisor of M) of azimuthal tracks, and a duty ratio pattern of the control pulses in a number L (L=M/N) of successive periods corresponding to one frame is different in each of a number K (where K is an integer equal to or larger than two) of successive frames.

Non-restoring division device

Abstract: PURPOSE: To execute division at high speed by providing an addition/subtraction circuit adding/subtracting a dividend and a divisor and a judgement register judging the positive/negative of the result of addition/subtraction and supplying a judged result to a dividend storage register and a second divisor selection circuit as a selection signal. CONSTITUTION: The dividend storage register 1 stores dividend data and divisor storage registers 2, 3 and 4 store onefold, twofold and threefold data of the divisor. A control circuit 5 generates control signals selecting onefold, twofold and threefold data of divisor data. A first divisor selection circuit 6 selects onefold, twofold and threefold divisor data of the divisor. The addition/ subtraction circuit 7 adds/subtracts the dividend and the divisor, and an operation result storage register 8 stores an addition/subtraction result. A quotient generation circuit 9 generates a part of the quotient and a quotient storage register 10 stores the quotient by division. A quotient subtraction circuit 11 subtracts the quotient which is stored when a subtraction result is negative. A judgement register 12 stores the judgement signal of positive/negative information.

Method for calculating frequency-dividing ratio in numeral phase-locked loop and apparatus thereof

Abstract: The method involves comparing (12) a reference oscillation (26) with a feedback oscillation (28), at a triggering event, to produce an error signal (30,32). The error signal is compared with a coarse threshold (36) and, if it compares unfavourably, a coarse divisor adjustment (40,18) is provided to a feedback divider (20). If the error signal compares favourably with the coarse threshold, it is compared with a fine threshold (38). If it compares unfavourably with the fine threshold, a fine divisor adjustment (42,18) is provided to the feedback divider.

A lower bound for $K_{X}L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension

Abstract: Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa(X)=0$ or 1, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.