On quadratic fields generated by polynomials

Let be a polynomial of degree d ≥ 2 without multiple roots. Under the assumption of the ABC-conjecture, an asymptotic formula for the number of distinct fields among for has recently been given by Cutter, Granville, and Tucker. We use bounds for character sums to obtain an unconditional lower bound on the number of such fields for . Received: 19 November 2007     

Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)We consider existence and qualitative properties of standing wave solutions to the nonlinear Schr?dinger equation with E being a critical frequency in the sense that inf . We verify that if the zero set of W − E has several isolated points x _i () near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, , a standing wave solution which concentrates simultaneously on , where is any given subset of . This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Linear Chromatic Bounds for a Subfamily of 3K 1-free Graphs

For an arbitrary class of graphs , there may not exist a function f such that , for every . When such a function exists, it is called a χ-binding function for . The problem of finding an optimal χ-binding function for the class of 3K ₁-free graphs is open. In this paper, we obtain linear χ-binding function for the class of {3K ₁, H}-free graphs, where H is one of the following graphs: , House graph and Kite graph. We first describe structures of these graphs and then derive χ-binding functions.     

Parabolic Raynaud bundles

Let X be an irreducible smooth projective curve defined over the field of complex numbers, a finite set of closed points and N ≥ 2 a fixed integer. For any pair , there exists a parabolic vector bundle on X, with parabolic structure over S and all parabolic weights in , that has the following property: Take any parabolic vector bundle of rank r on X whose parabolic points are contained in S, all the parabolic weights are in and the parabolic degree is d. Then is parabolically semistable if and only if there is no nonzero parabolic homomorphism from to .     

Degrees of difficulty of generalized r.e. separating classes

Important examples of classes of functions are the classes of sets (elements of ^ω 2) which separate a given pair of disjoint r.e. sets: . A wider class consists of the classes of functions f ∈ ^ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let denote the Medvedev degrees of those such that no m + 1 sets among A ₀,...,A _k-1 have a nonempty intersection. It is shown that each is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions is the greatest element of . If 2 ≤ l < k, then 0 _M is the only degree in which is below a member of . Each is densely ordered and has the splitting property and the same holds for the lattice it generates. The elements of are exactly the joins of elements of for . Supported by National Science Foundation grants DMS 0554841, 0532644 and 0652732.     

Regularity of the Bergman Projection on Forms and Plurisubharmonicity Conditions

Let be a smoothly bounded domain. Suppose Ω has a defining function, such that the sum of any q eigenvalues of its complex Hessian is non-negative. We show that this implies global regularity of the Bergman projection, B _j-1, and the -Neumann operator, N _j , acting on (0,j)-forms, for .Research of the first author was partially supported by a Rackham Fellowship.Research of the second author was partially supported by an NSF grant.     

Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, a nontrivial additive character, and a character on . Then ψ defines an Artin-Schreier sheaf on the affine line , and χ defines a Kummer sheaf on the n-dimensional torus . Let be a Laurent polynomial. It defines a k-morphism . In this paper, we calculate the weights of under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

The optimal statistical median of a convex set of arrays

We consider the following problem. A set of vectors is given. We want to find the convex combination such that the statistical median of z is maximum. In the application that we have in mind, are the historical return arrays of asset j and are the portfolio weights. Maximizing the median on a convex set of arrays is a continuous non-differentiable, non-concave optimization problem and it can be shown that the problem belongs to the APX-hard difficulty class. As a consequence, we are sure that no polynomial time algorithm can ever solve the model, unless P = NP. We propose an implicit enumeration algorithm, in which bounds on the objective function are calculated using continuous geometric properties of the median. Computational results are reported.     

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!     

Minimizing total completion time for UET tasks with release time and outtree precedence constraints

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n ²)-time algorithm for the problems , outtree and , outtree . In this note, we show that their algorithm admits an O(n log n)-time implementation.     

Derived McKay correspondence via pure-sheaf transforms

In most cases where it has been shown to exist the derived McKay correspondence can be written as a Fourier–Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in . We give a sufficient condition for to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of . Along the way we extract more geometrical meaning out of the Intersection Theorem and learn to compute θ-stable families of G-constellations and their direct transforms.     

On the optimum capacity of capacity expansion problems

In this paper we consider problems of the following type: Let E = { e ₁, e ₂,..., e _n } be a finite set and be a family of subsets of E. For each element e _i in E, c _i is a given capacity and _i is the cost of increasing capacity c _i by one unit. It is assumed that we can expand the capacity of each element in E so that the capacity of family can be expanded to a level r. For each r, let f (r) be the efficient function with respect to the capacity r of family , and be the cost function for expanding the capacity of family to r. The goal is to find the optimum capacity value r ^* and the corresponding expansion strategy so that the pure efficency function is the largest. Firstly, we show that this problem can be solved efficiently by figuring out a series of bottleneck capacity expansion problem defined by paper (Yang and Chen, Acta Math Sci 22:207–212, 2002) if f (r) is a piecewise linear function. Then we consider two variations and prove that these problems can be solved in polynomial time under some conditions. Finally the optimum capacity for maximum flow expansion problem is discussed. We tackle it by constructing an auxiliary network and transforming the problem into a maximum cost circulation problem on the auxiliary network.     

Closed polynomials and saturated subalgebras of polynomial algebras

The behavior of closed polynomials, i.e., polynomials such that the subalgebra is integrally closed in , is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras , i.e., subalgebras such that, for any , a generative polynomial of f is contained in A. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1587–1593, December, 2007.     

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.     

Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK]. Received: May 2007, Revision: October 2007, Accepted: October 2007     

On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.     

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

On the stability of linear systems with an exact constraint set

This paper deals with the stability of the intersection of a given set with the solution, , of a given linear system whose coefficients can be arbitrarily perturbed. In the optimization context, the fixed constraint set X can be the solution set of the (possibly nonlinear) system formed by all the exact constraints (e.g., the sign constraints), a discrete subset of (as or { 0,1} ⁿ , as it happens in integer or Boolean programming) as well as the intersection of both kind of sets. Conditions are given for the intersection to remain nonempty (or empty) under sufficiently small perturbations of the data. Research supported by Fondecyt Grant 1020(7020)-646. Research supported by DGES and FEDER, Grant BFM2002-04114-C02-01