The curve of ``Prym canonical' Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:

1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;

2) those divisors in the Gauss system parametrized by the canonical curve are reducible.

In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.

     

Approximating the number of integers free of large prime factors

Define to be the number of positive integers such that has no prime divisor larger than . We present a simple algorithm that approximates in floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the well-known approximation , where is Dickman's function.

     

General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the -exponential functions

and

as well as the Appell function

and the multivariate form of the partial theta function

     

Classification of extremal contractions from smooth fourfolds of -type

We investigate divisorial contractions of extremal rays from
smooth fourfolds. When the exceptional divisor is contracted to a curve, we prove that the divisor is a -bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve. Furthermore we determine the local analytic structure of the contraction.

     

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give

• a sufficient criterion for the property,
• a simple proof of F.J. Calderón-Moreno's theorem that free divisors have the property,
• a proof that divisors in dimension with only isolated quasi-homogeneous singularities have the property,
• an example of a nonfree divisor with nonisolated singularity having the property,
• an example of a divisor not having the property, and
• an algorithm to compute the V-filtration along a divisor up to a given order.

     

Remark on ``A problem of prescribing Gaussian curvature on " [Proc. Amer. Math. Soc. 129 (2001), no. 12, 3757-3758]

In this note, we remark on a 2001 paper of S. Goyal and V. Goyal. The main result of this work is that they used some elementary method to find a class of functions for which the solutions to

on can be obtained. We observe that this class of functions that they studied is actually the trivial one, i.e. the class of positive constant functions.

     

Sub- and superadditive properties of Euler's gamma function

Let and be real numbers. The inequality

holds for all positive real numbers if and only if . The reverse inequality is valid for all if and only if .

     

Bounding the number of cycles of O.D.E.s in

Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the differential equation in . The special case of an annulus is further considered, and the criteria are used to deduce sufficient conditions for the differential equation to have at most one cycle. A bound on the number of cycles on surfaces of higher connectivity is given by similar conditions.

     

A simple formula for an analogue of conditional Wiener integrals and its applications

Let denote the space of real-valued continuous functions on the interval and for a partition of , let be given by .

In this paper, with the conditioning function , we derive a simple formula for conditional expectations of functions defined on which is a probability space and a generalization of Wiener space. As applications of the formula, we evaluate the conditional expectation of functions of the form

for and derive a translation theorem for the conditional expectation of integrable functions defined on the space .

     

The arithmetic of certain cubic function fields

In this paper, we discuss the properties of curves of the form over a given field K of characteristic different from 3. If satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.

     

Heights of algebraic points lying on curves or hypersurfaces

Our first aim will be to give an explicit version of a generalization of the results of Zhang and Zagier on algebraic points with . Secondly, we will show that distinct algebraic points lying on a given curve of certain type can be distinguished in terms of some height functions. Thirdly, we will derive a bound for the number of points on such a curve whose heights are under a given bound and whose coordinates lie in a multiplicative group of given rank.

     

Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation

for a given function . We find its general solution assuming only continuous differentiability and local nonlinearity of . We also provide new results about the more general equation

for a given function . Previous uniqueness results required strong regularity assumptions on a particular solution . Here we weaken the assumptions on considerably and find all solutions under slightly stronger regularity assumptions on .

     

Hyperbolic conservation laws with stiff reaction terms of monostable type

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type

is studied. Solutions of Cauchy problems of the above equation with initial value are proved to converge, as , to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed . The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving 0$"> is found to originate from the behavior of traveling waves of the above system with viscosity regularization.

     

Gaussian curvature in the negative case

In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case.

In 1974, Kazdan and Warner considered the equation

on any compact two dimensional manifold with . They showed that there exists a number , such that the equation is solvable for every \alpha > \alpha_o$"> and it is not solvable for .

Then one may naturally ask:

Is the equation solvable for ?

In this paper, we answer the question affirmatively. We show that there exists at least one solution for .

     

Linear difference equations with transition points

Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation

where and have power series expansions of the form

with . Our results hold uniformly for in an infinite interval containing the transition point given by . As an illustration, we present an asymptotic expansion for the monic polynomials which are orthogonal with respect to the modified Jacobi weight , , where , -1$"> and is real analytic and strictly positive on .

     

Limiting weak-type behavior for singular integral and maximal operators

The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator in : for , ,

For the maximal operator , the corresponding result is

     

A new characterization of ultraspherical polynomials

We characterize the class of ultraspherical polynomials in between all symmetric orthogonal polynomials on via the special form of the representation of the derivatives by

     

On the number of solutions of an algebraic equation on the curve , and a consequence for o-minimal structures

We prove that every polynomial of degree has at most zeros on the curve . As a consequence we deduce that the existence of a uniform bound for the number of zeros of polynomials of a fixed degree on an analytic curve does not imply that this curve belongs to an o-minimal structure.

     

A unified approach to evaluation algorithms for multivariate polynomials

We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree in variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity , a nested multiplication algorithm with computational complexity and a ladder recurrence algorithm with computational complexity . These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity , a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities , and per point respectively for the evaluation of multivariate polynomials at several points.