The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

On the index of the Weierstrass semigroup of a pair of points on a curve

We obtain the exact formulas for the cardinality of the complement of the Weierstrass semigroup of a pair (p, q) of points on a curveC. Using these formulas we obtain lower bounds and upper bounds on the cardinalities of such sets. Moreover, considering examples, we show that these bounds are sharp.Partially supported by Korea Science and Engineering Foundation and by Global Analysis Research Center.     

Elements of the Lagrangian Whitney trick

We investigate the possibility of a Lagrangian Whitney trick, a process to remove a pair of intersection points of a self-transverse Lagrangian immersion by a homotopy through Lagrangian immersions. There is a model for which a Lagrangian Whitney trick with compact support works assuming the model satisfies an area-capacity condition. Reduction of more general cases to the model, not necessarily fulfilling the area-capacity requirement, is possible if the given pair of double points admits a suitable symplectic disc and a certain Maslov-Viterbo index is 1. We look into an example to see the actualities of the Maslov-Viterbo index and the area-capacity conditions. The authors would like to thank the anonymous referee whose suggestions improved remarkably the exposition of the paper. This work was supported by the grant no.R01-2000-000-00004-0 from Korea Science & Engineering Foundation.     

Minimal circumscribed Hermitian ellipsoid of Hartogs domains and an application to an extremal problem

In this paper we construct circumscribed Hermitian ellipsoids of Hartogs domains of least volume and as an application, we obtain the Carath′eodory extremal mappings between the Hartogs domains and the unit ball, and also give an explicit formula for calculating the extremal values.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Weierstrass Semigroups of a Pair of Points Whose First Nongaps are Three

We found all candidates for a Weierstrass semigroup at a pair of Weierstrass points whose first nongaps are three. We prove that such semigroups are actually Weierstrass semigroups by constructing examples.     

Existence of the non-primitive Weierstrass gap sequences on curves of genus 8

We show that for any possible Weierstrass gap sequence L on a non-singular curve of genus 8 with twice the smallest positive non-gap is less than the largest gap there exists a pointed non-singular curve (C, P) over an algebraically closed field of characteristic 0 such that the Weierstrass gap sequence at P is L. Combining this with the result in [6] we see that every possible Weierstrass gap sequence of genus 8 is attained by some pointed non-singular curve. *Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. **Partially supported by Grant-in-Aid for Scientific Research (17540030), Japan Society for the Promotion of Science.     

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPC ^g−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail. Research of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis (supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Weierstrass semigroups whose minimum positive integers are even

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3. Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. Received: 19 June 2006     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004     

Large deviations and importance sampling for a tandem network with slow-down

We consider a variant of the two-node tandem Jackson network where the upstream server reduces its service rate when the downstream queue exceeds some prespecified threshold. The rare event of interest is the overflow of the downstream queue. Based on a game/subsolution approach, we rigorously identify the exponential decay rate of the rare event probabilities and construct asymptotically optimal importance sampling schemes. Research of P. Dupuis supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003) and the Army Research Office (W911NF-05-1-0289). Research of K. Leder supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003). Research of H. Wang supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003).     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

Large regular graphs with no induced 2K 2

Letr be a positive integer. Considerr-regular graphs in which no induced subgraph on four vertices is an independent pair of edges. The numberv of vertices in such a graph does not exceed 5r/2; this proves a conjecture of Bermond. More generally, it is conjectured that ifv>2r, then the ratiov/r must be a rational number of the form 2+1/(2k). This is proved forv/r≥21/10. The extremal graphs and many other classes of these graphs are described and characterized. Research supported in part by the National Science Foundation under ISP 80110451. Research supported in part by the National Science Foundation under DMS-8401281. Research supported in part by the National Science Foundation under DMS-8504322, and by the Office of Naval Research under N00014-85K0570.     

Crossing families

Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two setsA andB of points in the plane are mutually avoiding if no line subtended by a pair of points inA intersects the convex hull ofB, and vice versa. We show that any set ofn points in general position contains a pair of mutually avoiding subsets each of size at least . As a consequence we show that such a set possesses a crossing family of size at least , and describe a fast algorithm for finding such a family.Research supported in part by DARPA grant N00014-89-J-1988, Air Force AFOSR-89-0271, NSF grant DMS-8606225, and an ONR graduate fellowship. Further, part of this work was conducted at and supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC8809648.     

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.     

Minimal pairs in initial segments of the recursively enumerable degrees

We show that for every r.e. Turing degreea>0, there is an r.e. degreeb<a which is not half of a minimal pair in the initial segment [0, a]. This work was partially supported by National Science Foundation Grant DMS88-00030 to Stob, the New Zealand Marsden Fund for Basic Research in Science Grant VIC-509 to Downey, and New Zealand-United States Cooperative Science Program Grant INT90-20558 from the National Science Foundation to both authors. The authors thank the referee for several helpful comments.     

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties. – Dedicated to the treasured memory of our coauthor, Paulo Henrique Viana