Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves

Abstract: This paper gives a very good systematic presentation of the equivalence between the algebraic function fields in one variable over the field $\bbfR$ of real numbers and the Klein surfaces. In section 1 Klein surfaces and morphisms between them are defined, and example as well as the basic facts about them are given. The double covering of a Klein surface and the quotient of a Riemann surface under an antianalytic involution is described, and it is noted that these two constructions are mutually inverse. Section 2 is devoted to the notion of a meromorphic function of a compact Klein surface. It is shown that the field of meromorphic functions of a compact Klein surface is an algebraic function field in one variable over $\bbfR$. Also there exists a functor of the category $\cal K$ of compact Klein surfaces to the category ${\cal F}\sb \bbfR$ of the algebraic function fields in one variable over $\bbfR$. An intensive study of the set $S(E\mid\bbfR)$ of proper valuation rings $V$ of $E\in{\cal F}\sb \bbfR$ with $V\supset\bbfR$ is the object of section 3. The main results of this section are:\par (a) The residue field of $V\in S(E\mid\bbfR)$ is $\bbfR$ iff $E$ admits some ordering with respect to which $V$ is convex.\par (b) The Riemann theorem about the dimension $\ell(L)$ of the space $L(D)$ associated to the divisor $D$ of $E\mid \bbfR$.\par With these notations it is proved in section 4 that $S(E\mid\bbfR)$ admits a unique structure of a Klein surface for which $p:S(E(\sqrt{- 1})(\bbfC)\to S(E\mid\bbfR)$ is a morphism of Klein surfaces and $M(S(E\mid\bbfR))=E$. Further it is shown that every compact Klein surface $S$ is isomorphic to $S(M(S)\mid\bbfR)$. Also: $S\mapsto M(S)$ and $E\mapsto S(E\mid\bbfR)$ give an equivalence between $\cal K$ and ${\cal F}\sb \bbfR$. Here the Klein surfaces with non empty boundary correspond to the formally real fields. Among a series of interesting comments and remarks we mention merely two:\par (i) There are several non homeomorphic curves with the same field of rational functions, but there is a unique one among them which is irreducible, compact, non-singular and affine.\par (ii) The Klein surface $S$ with empty boundary is orientable iff $M(S)$ contains $\bbfC$.\par This carefully written paper is very interesting and recommended even for specialists.

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.

Integrals of Borcherds forms

Abstract: In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $\Psi(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his construction is a vector valued modular form of weight $1-n/2$ for $SL_2(Z)$ which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers $c_\mu(-m)$, $m\ge0$. For example, the divisor of $\Psi(F)$ is the sum over $m>0$ and the coset parameter $\mu$ of $c_\mu(-m) Z_\mu(m)$ for certain rational quadratic divisors $Z_\mu(m)$ on the arithmetic quotient $X = \Gamma D$. In this paper, we give an explicit formula for the integral $\kappa(\Psi(F))$ of $-\log||\Psi(F)||^2$ over $X$, where $||.||^2$ is the Petersson norm. More precisely, this integral is given by a sum over $\mu$ and $m>0$ of quantities $c_\mu(-m) \kappa_\mu(m)$, where $\kappa_\mu(m)$ is the limit as $Im(\tau) -> \infty$ of the $m$th Fourier coefficient of the second term in the Laurent expansion at $s= n/2$ of a certain Eisenstein series $E(\tau,s)$ of weight $n/2 + 1$ attached to $V$. It is also shown, via the Siegel--Weil formula, that the value $E(\tau, n/2)$ of the Eisenstein series at this point is the generating function of the volumes of the divisors $Z_\mu(m)$ with respect to a suitable Kahler form. The possible role played by the quantity $\kappa(\Psi(F))$ in the Arakelov theory of the divisors $Z_\mu(m)$ on $X$ is explained in the last section.

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.

Gigantic pairs of minimal clones

Abstract: L. Szabo (1992) asked for the minimal number n=n(|A|) such that the clone of all operations on A can be generated as the join of n minimal clones. He showed, e.g., n(p)=2 for any prime p, and later G. Czedli (1998) proved that if k has a divisor /spl ges/5 then n(k)=2. In this paper, a pair (f, g) of operations is called gigantic if each of f and g generates a minimal clone and the set {f, g} generates the clone of all operations. First, we give a general theorem to characterize a gigantic pair. Then we show that n(k)=2 for every k which is not a power of 2.