Showing papers on "Divisor published in 2003"

Normal affine surfaces with C*-actions

Abstract: A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Bia{\l}ynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of such surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction in the case of a hyperbolic grading. As an apllication we determine the structure of singularities, of the orbits and the divisor class groups for such surfaces.

System and method for divisor threshold control in a modulation domain divider

Abstract: A modulation domain divider is disclosed that causes the divider output to be attenuated when the divisor input falls below the divisor threshold. Attenuation is accomplished by implementing the divider in the modulation domain and substituting an unmodulated signal for the normal modulated signal when the divisor is below the threshold value.

More etale covers of affine spaces in positive characteristic

Abstract: We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H. This improves our earlier result in math.AG/0207150, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.

Thresholds of the Divisor Methods

Abstract: We give an algorithm which permits calculating the maximum and minimum vote shares that allow a party to obtain h seats, that is, the threshold of exclusion and the threshold of representation. These have already been studied for some methods (such as d'Hondt or Sainte-Lague), and are here generalized to any divisor method, and to any number of seats. The thresholds depend on the size of the constituency, the number of parties running in the constituency, and the divisor method used. Finally, we give some consequences, including a characterization of the d'Hondt method.

On Dirichlet Series for Sums of Squares

Abstract: Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions σ k (n) and σ k 2 (n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f i and g i are completely multiplicative, then we have $$ \sum\limits_{n = 1}^\infty {\frac{{({f_{1*{g_1}}})\cdot({f_{2*{g_2}}})(n)}}{{{n^s}}}} = \frac{{{L_{{f_1}{f_2}}}(s){L_{{g_1}{g_2}}}(s){L_{{f_1}{g_2}}}(s){L_{{g_1}{f_2}}}(s)}}{{{L_{{f_1}{f_2}{g_1}{g_2}}}(2s)}} $$ where \( {L_f}(s): = \sum olimits_{n = 1}^\infty {f(n){n^{ - s}}} \) is the Dirichlet series corresponding to f. Let r N (n) be the number of solutions of x 1 2 + … + x N 2 = n and r 2, p (n) be the number of solutions of x 2 + Py 2 = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ (s) and Dirichlet L-functions, for the generating functions of r N (n), r N 2 (n), r 2, p (n) and r 2, p (n)2 for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.

Toric Fano varieties and birational morphisms

Abstract: In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.

New bounds for automorphic L-functions

Abstract: This dissertation contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation \pi of GL(m) over a number field. The approximation involves a smooth truncation of the Dirichlet series L(s,\pi) and L(s,\tilde{\pi}) after about \sqrt{C} terms, where C denotes the analytic conductor (of \pi and \tilde{\pi} at the central point) introduced by Iwaniec and Sarnak. We investigate the decay rate of the cutoff function and its derivatives. We also see that the truncation can be made uniformly explicit at the cost of an error term. The results extend to products of central values. We establish, via the Hardy–Littlewood circle method, a nontrivial bound on shifted convolution sums of Fourier coefficients coming from classical holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. We achieve polynomial uniformity in all the parameters of the cusp forms by carefully estimating the Bessel functions that enter the analysis. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms. We also study the shifted convolution sums via the Sarnak–Selberg spectral method. For holomorphic cusp forms this approach detects optimal cancellation over any totally real number field. For Maass cusp forms the method is burdened with complicated integral transforms. We succeed in inverting the simplest of these transforms whose kernel is built up of Gauss hypergeometric functions.

Some characterizations of specially multiplicative functions

Abstract: A multiplicative function f is said to be specially multiplicative if there is a completely multiplicative function fA such that f(m)f(n)=∑d|(m,n)f(mn/d2)fA(d) for all m and n. For example, the divisor functions and Ramanujan's τ-function are specially multiplicative functions. Some characterizations of specially multiplicative functions are given in the literature. In this paper, we provide some further characterizations of specially multiplicative functions.

Receiver architecture eliminating static and dynamic DC offset errors

Abstract: The present invention provides a receiver frontend that eliminates static and dynamic DC errors and has improved second order intermodulation distortion (IMD2) performance. The receiver frontend includes a first mixer that multiplies a received signal and a first local oscillator (LO) signal to produce an intermediate frequency (IF) signal. A second mixer multiplies the IF signal and a second LO signal to produce an output signal. A first divider circuit divides a reference signal from a reference oscillator by a first divisor N to produce the first LO signal, and a second divider circuit divides the reference signal by a second divisor M to produce the second LO signal. Preferably, the first and second divisors N and M are each integers greater than one (1), and the second divisor M is not an integer multiple of the first divisor N.

Addendum to "Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees," American Journal of Mathematics 119 (1997), 1139-1172

Abstract: This addendum uses Bloch's original technique of ordinary differential equations to circumvent a difficulty in the proof of Lemma 2 in the paper in the title. This technique at the same time yields the following stronger Nevanlinna's Second Main Theorem with truncation at an order given explicitly by the Chern class of the divisor. For an ample divisor D in an abelian variety A of complex dimension n and for any holomorphic map f : C → A whose image is not contained in any translate of D , the characteristic function of f for D is dominated by the counting function of f for D truncated at order k n plus an error term of logarithmic order of the characteristic function, where k n is inductively given by k 0 = 0, k 1 = 1, and k l+1 = k l + 3 n -l-1 (4( k l + 1)) l D n for 1 ≤ l n .

Revisiting SRT quotient digit selection

Abstract: The quotient digit selection in the SRT division algorithm is based on a few most significant bits of the remainder and divisor, where the remainder is usually represented in a redundant representation. The number of leading bits needed depends on the quotient radix and digit set, and is usually found by an extensive search, to assure that the next quotient digit can be chosen as valid for all points (remainder, divisor) in a set defined by the truncated remainder and divisor, i.e., an "uncertainty rectangle". We present expressions for the number of bits needed for the truncated remainder and divisor, thus eliminating the need for a search through the truncation parameter space for validation. We also present simple algorithms to properly map truncated negative divisors and remainders into nonnegative values, allowing the quotient selection function only to be defined on the smaller domain of nonnegative values.

Numerical criteria for divisors on $\M_{g}$ to be ample

Abstract: The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional components of this stratification are smooth rational curves (whose numerical equivalence classes are) called F-curves. The F-conjecture asserts that a divisor on $\M_{g,n}$ is ample if and only if it positively intersects the $F-$curves. In this paper the F-conjecture on $\M_{g,n}$ is reduced to showing that certain divisors in $\M_{0,N}$ for $N \leq g+n$ are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. As an application of the reduction, numerical criteria are given which if satisfied by a divisor $D$ on $\M_g$, show that $D$ is ample. Additionally, an algorithm is described to check that a given divisor is ample. Using a computer program called Nef Wizard, written by Daniel Krashen (this http URL), one can use the criteria and the algorithm to verify the conjecture for low genus. This is done on $\M_g$ for $g \le 24$, more than doubling the known cases of the conjecture and showing it is true for the first genus such that $\M_g$ is known to be of general type.

Tight upper bounds on the minimum precision required of the divisor and the partial remainder in high-radix division

Abstract: Digit-recurrence binary dividers are sped up via two complementary methods: keeping the partial remainder in redundant form and selecting the quotient digits in a radix higher than 2. Use of a redundant partial remainder replaces the standard addition in each cycle by a carry-free addition, thus making the cycles shorter. Deriving the quotient in high radix reduces the number of cycles (by a factor of about h for radix 2/sup h/). To make the redundant partial remainder scheme work, quotient digits must be chosen from a redundant set, such as [-2, 2] in radix 4. The redundancy provides some tolerance to imprecision so that the quotient digits can be selected based on examining truncated versions of the partial remainder and divisor. No closed form formula for the required precision in the partial remainder and divisor, as a function of the quotient digit set and the range of the partial remainder, is known. We establish tight upper bounds on the required precision for the partial remainder and divisor. The bounds are tight in the sense that each is only one bit over a well-known simple lower bound. We also discuss the implications of these bounds for the quotient digit selection process.

Fourier-Mukai transforms and canonical divisors

Abstract: Let $X$ be a smooth projective variety. We study a relationship between the derived category of $X$ and that of a canonical divisor. As an application, we will study Fourier-Mukai transforms when $\kappa (X)=dim X-1$.

A Generalization of Synthetic Division and A General Theorem of Division of Polynomials

Abstract: 01 1 1 ) ( a x a x a x a x f n n n n + + + + = − − L by a binomial of c x x g − = ) ( , without mentioning if this classical method can be applied when the divisor is a polynomial of degree being higher than 1, and some further explicitly stated that it is not applicable to such a divisor. For example, Larson, Hostetler, and Edwards claimed, “synthetic division works only for divisors of the form k x − . You cannot use synthetic division to divide a polynomial by a quadratic such as 3

Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere

Abstract: This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the results of Arinkin and Lysenko to the case of arbitrary number $n$ of points. First, we construct separated Darboux coordinated in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Sklyanin formulas. In the second part of the paper, we construct Drinfeld's compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlev´e-VI system.

Orbits of Curves on Certain K3 Surfaces

Abstract: In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in \(\mathbb{P}^{2} { \times }\mathbb{P}^{2} \) defined over \(\mathbb{C}\) and with Picard number 3. We describe the group of automorphisms \(\mathcal{A} = Aut (V / {\mathbb{C}})\) on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity \(N_{\mathcal{A}{\text{(}}C_0 {\text{)}}} (t) = \# \{ C \in \mathcal{A}{\text{(}}C_0 {\text{)}}\;{\text{:}}\;C \cdot D < t\} \). We show that the limit $$\mathop {\lim }\limits_{t \to \infty } \frac{{\log N_{\mathcal{A}(C)} (t)}}{{\log t}} = \alpha $$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

N -Covers of hyperelliptic curves

Abstract: For a hyperelliptic curve C of genus g with a divisor class of order n=g+1, we shall consider an associated covering collection of curves D$_\delta$, each of genus g$^2$. We describe, up to isogeny, the Jacobian of each D$_\delta$ via a map from D$_\delta$ to C, and two independent maps from D$_\delta$ to a curve of genus g(g-1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.

Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture

Abstract: We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give a counterexample to the Slope Conjecture. The computation relies on the fact that this divisor has four different incarnations as a geometric subvariety of the moduli space of curves, one of them as a higher rank Brill-Noether divisor consisting of curves with an exceptional rank 2 vector bundle. As an application we describe the birational nature of the moduli space of n-pointed curves of genus 10, for all n. We also show that on M_{11} there are effective divisors of minimal slope and having large Iitaka dimension. This seems to contradict the belief that on M_g the classical Brill-Noether divisors are essentially the only ones of slope 6+12/(g+1).

Minimal algebraic surfaces of general type with $c^2_1 = 3, p_g = 1 \text{ and } q = 0$, which have non-trivial 3-torsion divisors

Abstract: We shall give a concrete description of minimal algebraic surfaces $X$’s defined over $\mathbb{C}$ of general type with the first chern number 3, the geometric genus 1 and the irregularity 0, which have non-trivial 3-torsion divisors. Namely, we shall show that the fundamental group is isomorphic to $\mathbb{Z}/3$, and that the canonical model of the universal cover is a complete intersection in $\mathbb{P}^{4}$ of type (3, 3).

Methods and apparatus for a system clock divider

Abstract: A power-saving clock divider scheme is cost-effective, flexible, jitterless, and allows the user to keep track of time. In general, the clock divider selectively operates in a normal mode and one or more divide modes, wherein the divide modes provide a clock frequency that is a fraction of the normal clock frequency by a divisor value that is specified in a user-accessible divider register. Lower divisor values (e.g., 2, 4, 8, etc.) are preferably used for performance tuning, while large divisor values (e.g., 1024, 2048, and 4096) are preferably used for power saving.

On a sequence related to the Josephus problem

Abstract: In this short note, we show that an integer sequence defined on the minimum of differences between divisor complements of its partial products is connected with the Josephus problem (q=3).

Fitting for switchover of fluid path, has flow divisor with by-passes, and rotary disk-shaped steering elements that respectively sealing the axial face of the corresponding by-pass

Abstract: A flow divisor (7) has by-passes (17,18), with rotary disk-shaped steering elements (12,13) respectively sealing the axial face of the corresponding by-pass.

Prescaled integer division

Abstract: We describe a high radix integer division algorithm where the divisor is prescaled and the quotient is postscaled without modifying the dividend to obtain an identity N=Q/sup *//spl times/D+R/sup */ with the quotient Q/sup */ differing from the desired integer quotient Q only in its lowest order high radix digit. Here the "oversized" partial remainder R/sup */ is bounded by the scaled divisor with at most one additional high radix digit selection needed to reduce the partial remainder and augment the quotient to obtain the desired integer division result N=Q/spl times/D+R with 0/spl les/R/spl les/D-1. We present a high radix multiplicative version of this algorithm where a k/spl times/p digit base /spl beta/ rectangular aspect ratio multiplier allows quotient digit selection in radix /spl beta//sup k-1/ with a cost of only one k/spl times/p digit multiply per high radix digit, plus the fixed pre- and post-scaling operation costs. We also present a Booth radix 4 additive version of this algorithm where appropriately compressed representation of the partial remainder with Booth digits {-2, -1, 0, 1, 2} allows successive quotient digit selection from the leading partial remainder digit without the iterative table lookups required in SRT division.

Division method and division device

Abstract: PROBLEM TO BE SOLVED: To provide a division device for which arithmetic accuracy is improved and a circuit scale is reduced. SOLUTION: A divisor is divided into a mantissa part and an exponent part, division is performed at the mantissa part and then only the bit shift of a power of two is performed by a bit shift circuit 5 for the calculation of the exponent part. By contriving the division of the divisor, the calculation of the mantissa part is performed by making the divisor be in common as much as possible in a division ROM 1 and a linear arithmetic interpolation circuit 2 as well.

Design of a cycle-efficient 64-b/32-b integer divisor using a table-sharing algorithm

Abstract: In new generations of microprocessors, the superscalar architecture is widely adopted to increase the number of instructions executed in one cycle. The division instruction among all of the instructions needs more cycles than the rest, e.g., addition and multiplication. This makes the division instruction an important cycles-per-instruction figure for modern microprocessors. In this paper, a radix-16/8/4/2 divisor is proposed, which uses a variety of techniques, including operand scaling, table partitioning, and, particularly, table sharing, to increase performance without the cost of increasing complexity. A physical chip using the proposed method is implemented by 0.35-/spl mu/m single poly four metal (1P4M) CMOS technology. The testing measurement shows that the chip can execute signed 64-b/32-b integer division between 3-13 cycles with a 80-MHz operating clock.

Inverse element calculation device and method, and rsa key pair production apparatus and method

Abstract: PROBLEM TO BE SOLVED: To provide an inverse element calculation unit in which the division of many precisions is not used, of which memory capacity is reduced, and which determines an inverse element on a residue class ring in which the value of a divisor is even. SOLUTION: The inverse element calculation unit consists of: the divisor unit which acquires M and t which satisfies N=M×2 t (M is the odd number), and a first inverse element calculation unit which calculates the inverse element on the residue class ring which makes the divisor as the acquired value M, for the value of the divisor whose number is even; and a second inverse element calculation unit which calculates the inverse element on the residue class ring which makes 2 t as the divisor, and a synthetic calculation unit which calculates the inverse element on the residue class ring which makes N as the divisor from a value acquired with each inverse element calculation unit, for the acquired t. Thus, the inverse element calculation is calculated by controlling every calculation units. COPYRIGHT: (C)2004,JPO

System, method, and apparatus for division coupled with truncation of signed binary numbers

Abstract: A system, method, and apparatus for efficient rounding of signed numbers is presented herein. If the divisor is positive, the dividend is added to one half of the magnitude of the divisor. If the divisor is negative, the complement of the dividend is added to one half of the magnitude of the divisor. If the dividend is negative, and the divisor is also negative, one is added to the sum of the inverted dividend and one-half of the magnitude of the divisor. If the dividend is negative and the divisor is positive, one is subtracted from the sum of the dividend and one-half the magnitude of the divisor. The result is then right shifted x times. If the signs of the divisor and dividend are different, a most-significant-bit(sign-bit) of the result is shifted in as the most significant bit during each right shift. Otherwise, a “0” is shifted in.

On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity

Abstract: An interval [a, a + d] of natural numbers verifies the property of no coprimeness if and only if every element a + 1, a + 2,..., a + d - 1 has a common prime divisor with extremity a or a + d. We show the set of such a and the set of such d are recursive. The computation of the first d leads to rise a lot of open problems.