Showing papers on "Divisor published in 2000"

Criteria for sigma-ampleness

Abstract: In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism of a projective scheme X. Many open questions regarding $\sigma$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $\sigma$-ample. As a consequence, we show right and left $\sigma$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.

On convex polygons of maximal width

Abstract: In this paper we consider the problem of finding the n-sided ( $n\geq 3$ ) polygons of diameter 1 which have the largest possible width w n . We prove that $w_4=w_3= {\sqrt 3 \over 2}$ and, in general, $w_n \leq \cos {\pi \over 2n}$ . Equality holds if n has an odd divisor greater than 1 and in this case a polygon $\cal P$ is extremal if and only if it has equal sides and it is inscribed in a Reuleaux polygon of constant width 1, such that the vertices of the Reuleaux polygon are also vertices of $\cal P$ .

Dissecting a Sieve to Cut Its Need for Space

Abstract: We describe a “dissected” sieving algorithm which enumerates primes in the interval [x 1, x 2], using \(O(x_{2}^{1/3})\) bits of memory and using \(O(x_{2} -- x_{1} + x^{1/3}_{2}\) arithmetic operations on numbers of \(O(\rm ln \it x_{2})\) bits. This algorithm is based on a recent algorithm of Atkin and Bernstein [1], modified using ideas developed by Voronoi for analyzing the Dirichlet divisor problem [20]. We give timing results which show our algorithm has roughly the expected running time.

On Numerically Effective Log Canonical Divisors

Abstract: Let $(X,\Delta)$ be a 4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisor $K_X+\Delta$ is semi-ample, if it is nef (numerically effective) and the Iitaka dimension $\kappa(X,K_X+\Delta)$ is strictly positive. For the proof, we use Fujino's abundance theorem for semi log canonical threefolds.

Methods and apparatus for performing pipelined SRT division

Abstract: An SRT division unit for performing a novel SRT division algorithm is presented. The novel SRT division algorithm comprises a method for performing SRT division using a radix r. As one skilled in the art will appreciate, the radix r dictates the number of quotient-bits k generated during a single iteration. The relationship between radix r and the number of quotient-bits k generated in a single iteration is r=2 k . The number of iterations needed to determine all quotient-digits is N, such that N=54/k for a 64 bit floating point value. In accordance with one embodiment of the present invention, the SRT division unit generates a scaling factor M, which comprises scaling sub-factors M1 and M2 according to the relationship M=r*M1+M2. Next, the division unit generates a scaled divisor Y by multiplying a divisor DR by scaling factor M, such that said scaled divisor Y=DR*M=r(DR*M1)+DR*M2. In addition, the division unit generates partial remainder values w[00] and w[0] by muliplying a dividend DD by scaling sub-factor M1 and scaling factor M, respectively. Partial remainder value w[00]=DD*M1, and partial remainder value w[0]=DD*M=r(DD*M1)+DD*M2. Scaled divisor Y and partial remainders w[0] and w[00] then are used to generate quotient-digits and additional partial remainders. Accordingly, the division unit performs iterations j which generate quotient-digits according to the formula q[j]=SEL(r 2 *w msb [j−2], q[j−1]). Also, the iterations generate additional partial remainders w[j] according to the formula w[j]=rw[j−1]−q[j−1]*Y. N iterations are performed, generating all quotient-digits for the division operation.

Toric Calabi-Yau Fourfolds Duality Between N=1 Theories and Divisors that Contribute to the Superpotential *

Abstract: We study issues related to F-theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi–Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data for the heterotic theory from the polyhedron. We give a procedure for constructing examples with given gauge groups and describe some of these examples in detail. Interesting features arise when the local pieces are fitted into a global manifold. An important issue is how to compute the superpotential explicitly. Witten has shown that the condition for a divisor to contribute to the superpotential is that it have arithmetic genus 1. Divisors associated with the short roots of non-simply laced gauge groups do not always satisfy this condition while the divisors associated to all other roots do. For such a ‘dissident’ divisor we distinguish cases for which χ(OD) > 1 corresponding to an X that is not general in moduli (in the toric case this corresponds to the existence of non-toric parameters). In these cases the ‘dissident’ divisor D does not remain an effective divisor for general complex structure. If however χ(OD) ≤ 0, then the divisor is general in moduli and there is a genuine instability.

Toric varieties whose blow-up at a point is Fano

Abstract: We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a $T$-fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to $\PP^n$ or to the blow-up of $\PP^n$ along a $\PP^{n-2}$. As expected, such results are proved using toric Mori theory due to Reid.

Large-area, active-backlight display

Abstract: A low-cost, large-area display system having a backlight in segments each positioned to illuminate a subfield of M rows of a fast supertwisted-nematic (STN) display of N rows. Fields of Q+1 subfields are addressed by the method known as Active Addressing using orthogonal waveforms of period MT/N where T is the frame time. A subfield is addressed for Q+1 periods MT/N of the row waveforms and illuminated during the last one. Fast STNs allow Q to be small leading to a small effective multiplex ratio with improved contrast and horizontal viewing-angle range. A few additional leading and trailing rows may be addressed to overcome vertical parallax. The row drivers are periodically connected by switches that simply ground un-addressed rows. With Q+1 also a divisor of N, subfield contributions to the column waveforms can be calculated once and used Q+1 times in each frame. For example, N=240, M=16, L=4 and Q=2 provide an effective multiplex ratio of 57 and allow at least 2.2 msec for pixels to turn on when the frame rate 1/T is 60 Hz. The viewing-angle range can also be expanded by moving subfields a few rows in the scan direction and advancing the integration time to equalize the brightness of pixels illuminated by the next segment. For example, if the turn-off time is 0.76 T and the integration time is shortened to 0.33 MT/N, equalization is possible with turn-on times as large as ¾ of the turn-off time without decreasing pixel transmittance by more than 50%. Dual-scan configurations using N=240, for example, can display VGA or 480 p formats.

Method and apparatus for 50% duty-cycle programmable divided-down clock with even and odd divisor rates

Abstract: A 50% duty-cycle divided-down clock with selectable divisor rates A simple architecture comprised of two n-bit counters, a state machine, 2 toggle flip-flops, and two 2-to-1 muxes is used to allow an input clock signal to be divided down by any divisor rate up to 2 n

Encoding method and device, decoding method and device, and systems using them

Abstract: In order to encode an original sequence of binary data (u), a first padding operation (508) is performed, supplementing the original sequence (u) so that the supplemented sequence (u) is divisible by a first divisor polynomial; a first recursive convolutional encoding operation (508) is performed, using the first divisor polynomial, encoding the supplemented original sequence (u); an interleaving operation (506) is performed, permuting the binary data in the original sequence (u) by means of a specific permutation, so as to obtain an interleaved sequence (u*); a second padding operation (510) is performed, supplementing the interleaved sequence (u*) so that the supplemented interleaved sequence (u*) is divisible by a second divisor polynomial (g2); and a second recursive convolutional encoding operation (510) is performed, using the second divisor polynomial, encoding the supplemented interleaved sequence (u*).

Equations Involving Arithmetic Functions of Factorials

Abstract: For any positive integer k let (k); (k); and (k) be the Euler function of k, the divisor sum function of k, and the number of divisors of k, respectively. Let f be any of the functions ; , or . In this note, we show that if a is any positive real number then the diophantine equation f(n!) = am! has only nitely many solutions ( m;n). We also nd all solutions of the above equation when a = 1.

Divisors on varieties over valuation domains

Abstract: LetX be a separated integral normal scheme of finite type over the valuation ringO υ. It is shown that the setD coh(X) of coherent fractionaryO X -ideals $$\mathcal{J} \subseteq \underline {K\left( \mathcal{X} \right)} $$ satisfying the relation $${\mathcal{J}} = \hat {\mathcal{J}} : = ({\mathcal{O}}_{\mathcal{X}} :({\mathcal{O}}_{\mathcal{X}} :{\mathcal{J}}))$$ —the so-called divisorialO X -ideals—forms a group with the composition law $$\left( {\mathcal{I}\mathcal{J}} \right) \mapsto \widehat{\mathcal{I}\mathcal{J}}$$ . This group posesses a natural embedding $$div : \mathcal{D}_{coh} \left( \mathcal{X} \right) \to Div\left( \mathcal{X} \right) \oplus \prod\limits_{v \in V} {v\left( {K\left( \mathcal{X} \right)} \right)} $$ , where Div(X) denotes the group of Weil divisors of the generic fibreX of $$\mathcal{X}\left| {Spec\left( {\mathcal{O}_ u } \right)} \right.$$ , and V is a set of valuations ofK X determined by a subset of the generic points of the fibres $$\mathcal{X} \times _{\mathcal{O}_ u } \mathcal{K}\left( \mathcal{P} \right),\mathcal{P} \in Spec\left( {\mathcal{O}_ u } \right)\backslash 0$$ . The image Div(X) of div is proved to satisfy $$Div(\mathcal{X}) = Div(\mathcal{X}) \oplus Ver(\mathcal{X})$$ with a subgroup $$Ver(\mathcal{X}) \subseteq \prod {_{v \in V} } v\left( {K\left( \mathcal{X} \right)} \right)$$ . The structure of Ver(X) is determined provided thatX satisfies additional conditions—for example, ifX is projective over Spec(O v ). These facts are deduced from general results on the semigroupD coh(X) of coherent divisorialO X -ideals on an integral schemeX: A criterion forD coh(X) to be a group based on the notion of so-called Pruferv-multiplication rings, and a valuation theoretic description of this group using valuations of K(X) naturally associated toX. The considerations leading to these results show thatD coh(X) can be understood as an ideal theoretic generalization of the group of Weil divisors on a normal noetherian scheme. Following this idea a criterion forD coh(X),X a separated integral normalO v -scheme of finite type, to be equal to the group of Cartier divisors onX is given. The criterion is obtained by showing that for any pointx on such a scheme the local generalized Weil divisor groupsD coh(Spec(O,x)) exist and by analyzing the structure of these groups.

Classification of Okamoto-Painlev\'e Pairs

Abstract: In this paper, we introduce the notion of an Okamoto-Painlev\'e pair (S, Y) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlev\'e equations introduced by K. Okamoto give examples of Okamoto-Painleve pairs, we find a new example of Okamoto-Painlev\'e pairs not listed in \cite{Oka}. We will give the complete classification of Okamoto-Painlev\'e pairs.

Partially-synchronous high-speed counter circuits

Abstract: Partially-synchronous and non-integer integrated circuit counters for dividing a high-speed reference clock signal with a selectable divisor have been provided. The circuits use a high-speed synchronous counter that cycles between the use of a selectable and a fixed divisor, to give the counter circuit a selectable overall division ratio. The partially-synchronous counter circuit uses asynchronous dividers to complete the division process and to minimize power consumption. A non-integer counter circuit is provided that includes a edge select mechanism to reduce power consumption in the division process. Examples are presented with specific number of stages, and corresponding divisors and divisor ranges. Method for implementing the above-mentioned partially-synchronous and non-integer counter circuits have also been provided.

Almost periodic distributions in tube domains

Abstract: The notion of an almost periodic (a.p.) distribution in a tube domain TG in Cn is introduced. Properties of such distributions are investigated; some of them are similar to the properties of ordinary a.p. functions on the axis. In these terms, the notion of an a.p. divisor in TG is introduced, and conditions for the existence of the density of such a divisor in TG′, {\(G' \Subset G\)}, are found. Bibliography: 12 titles.

Embeddings for 3-dimensional CR-manifolds

Abstract: We consider the problem of projectively embedding strictly pseudoconcave surfaces, X_ containing a positive divisor, Z and affinely embedding its 3-dimensional, strictly pseudoconvex boundary, \( M = - bX\_ \) We show that embeddability of M in affine space is equivalent to the embeddability of X_ or of appropriate neighborhoods of Z inside X_ in projective space. Under the cohomological hypotheses: \( H_{comp}^2\left( {{X_ - },\left( - \right)} \right) = 0 \) And \( {H^1}\left( {Z,{N_Z}} \right) = 0 \) these embedding properties are shown to be preserved under convergence of the complex structures in the C ∞-topology.

The k-free Divisor Problem

Abstract: Let \(\) be the number of k-free divisions of n, and let \(\) be the counting function of \(\). We improve on the known estimates for the error term in the asymptotic formula for \(\) under the assumption of the Riemann Hypothesis. We also obtain an unconditional asymptotic formula for \(\), for small y.

Cell-based implementation of radix-4/2 64b dividend 32b divisor signed integer divider using the COMPASS cell library

Abstract: A high-speed 64b/32b integer divider using the digit-recurrence division method and the on-the-fly conversion algorithm, is presented. A fast normaliser is used as the preprocessor of the proposed integer divider. To reduce maximum division time, the proposed divider uses radix-4/2 division, instead of the traditional radix-2 division. On-the-fly quotient adjustment is also realised in the converter module of the divider. The entire design is written in the Verilog hardware description language using the COMPASS 0.6 /spl mu/m 1P3M cell library (V3.0), and then synthesised by SYNOPSYS. Finally a real chip is fabricated and fully tested. The test results are very impressive. A performance evaluation of a 128b/64b signed integer divider using the same design methodology is also included in this study.

Real divisors on real curves

Abstract: Let C be an affine or projective smooth real algebraic curve, having a non-empty real part. Then every divisor E on C , which is linearly equivalent to its conjugate E^c , is also equivalent to a divisor supported on a set of real points of C .

The theory of quasi-divisors on cartesian products

Abstract: In this paper we study, using r-ideal systems, how some properties of the directed groups Gi,i = 1,2, can be transferred into their cartesian product G, and vice-versa. In particular, beginning from the structures (G i , r i ) and (G2,r2), we construct the r\ ®r2ideal system in G and we prove that (Gt,r{) are rf-Priifer groups, i = 1,2, if and only if (G, n r2) is an r\ (g)r2-Priifer group. Our main result is that the group G admits a theory of divisors, theory of quasi-divisors or strong theory of quasi-divisors, if and only if the groups Gi,i = 1,2, admit such a theory, respectively Finally, when the groups Gi,G2 and G admit theories of quasi-divisors, we investigate the relation between their corresponding Lorenzen t-groups and divisor class groups.

Worst-Case Analysis of an Algorithm for Computing the Greatest Common Divisor of n Inputs

Abstract: We give an exact worst-case analysis of an algorithm for computing the greatest common divisor of n inputs. The algorithm is extracted from a 1995 algorithm of de Rooij for fixed-base exponentiation.

Equidistribution of divisors for sequences of holomorphic curves

Abstract: We study holomorphic curves in ann-dimensional complex manifold on which a family of divisors parametrized by anm-dimensional compact complex manifold is given. If, for a given sequence of such curves, their areas (in the induced metric) monotonically tend to infinity, then for every divisor one can define adefect characterizing the deviation of the frequency at which this sequence intersects the divisor from the average frequency (over the set of all divisors). It turns out that, as well as in the classical multidimensional case, the set of divisors with positive defect is very rare. (We estimate how rare it is.) Moreover, the defect of almost all divisors belonging to a linear subsystem is equal to the mean value of the defect over the subsystem, and for all divisors in the subsystem (without any exception) the defect is not less than this mean value.

Limits and power of the simplest uniform and self-stabilizing phase clock algorithm

Abstract: In this paper, the phase clock algorithm which stabilizes on general graphs is studied on anonymous rings. The authors showed that this K-clock algorithm works with K/spl ges/2D, where D is the diameter of the graph. We prove that this algorithm works on unidirectional (bidirectional) rings iff K satisfies K>K'K/K'+n(2K>2K'-K/K'+n, respectively) where K' is the greatest divisor of K (K'/spl ne/K) and n is the size of the ring. From this characterization, we show that any ring stabilizes with some K<2D if K is odd. We also prove that, if K is prime, unidirectional and bidirectional rings stabilize with K<2[n/2]/spl sime/D and K<2[n/3]/spl sime/4D/3, respectively. Finally, we generalize the algorithm to synchronize any ring with any clock value.

Counting Prime Divisors on Elliptic Curves and Multiplication in Finite Fields

Abstract: Let K/F q be an elliptic function field. For every natural number n we determine the number of prime divisors of degree n of K/F q which lie in a given divisor class of K.

On the ingham divisor problem

Abstract: The main result of this paper is the following theorem. Suppose thatτ(n) = ∑ d|n l and the arithmetical functionF satisfies the following conditions: Then there exist constantsA 1,A 2, andA 3 such that for any fixed \g3\s>0 the following relation holds: $$\sum\limits_{n \leqslant x} {r(n)} r(n + 1)F(n) = A_1 xln^{\text{2}} x + A_{\text{2}} xlnx + A_{\text{3}} x + O(x^{5/6 + \varepsilon } + x^{1 - \alpha /6 + \varepsilon } ), x \to \infty .$$ . Moreover, if for any primep the inequality \vbf(p)\vb\s<1 holds and the functionF is strongly multiplicative, thenA 1\s>0.

An Inequality for the Determinant of the GCD Matrix and the GCUD Matrix

Abstract: Let S = {x_1, x_2,..., x_n } be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S**) ≥ det(S), where the equality holds if and Only if (S** ) = (S).

Arithmetic circuit for euclidean mutual division

Abstract: PROBLEM TO BE SOLVED: To reduce the circuit scale of a Euclidean mutual division arithmetic circuit for finding out an error position polynomial and an error evaluation polynomial necessary for a process for correcting an error by using a reed- solomon code. SOLUTION: In the arithmetic circuit, each of a 1st register group 101 for storing the coefficients of a divident polynomial and a 2nd register group 102 for storing the coefficients of a divisor polynomial consists of plural registers and a value stored in each register can be shifted to its upper register. The divi-dent polynomial stored in the 1st register group 101 is divided by the divisor polynomial stored in the 2nd register group 102, its remainder is shifted to the 2nd register group 102 and the coeffi-cients of the divisor polynomial are shitted to the 1st register group 101.

Harmonic reduction in a receiver using a variable mark-to-space ratio in a demodulator oscillator

Abstract: A receiver 20 includes two downconversion stages 23 and 24, 25 which are provided with local oscillator signals originating from a single local oscillator 27 A frequency divider 28 receives the signal provided by the local oscillator 27, and provides to the mixers 24, 25 a signal having a frequency equal to an integer divisor of the frequency of the local oscillator signal provided to the mixer 23 The mark-to-space ratio of the signal provided by the frequency divider 28 is arranged to adopt a value which minimises the values of either the (N+1) or the (N-1) harmonic, dependent on whether the receiver 29 is a low or high side injection receiver respectively, where N is the divisor ratio of the frequency divider