On Lang's Conjecture for Surfaces of General Type

Abstract

Let m be a nonnegative integer. We explicitly bound the number of rational curves with arithmetic genus no greater than m on a surface of general type by geometric invariants of the surface. We also briefly discuss the possibility of bounding all rational curves lying on a surface of general type in ?³.     

Differential Equations for Singular Vectors of sp(2n)

ABSTRACT

Given a weight λ of sp(2n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we find a family of exact solutions of the system in a certain space of power series. The polynomial solutions correspond to the singular vectors in the Verma module. In particular, we find the explicit expression of a singular vector corresponding to the single condition that ? λ,α? + ht α is a nonnegative integer for some positive root α, whose existence was proven by Jantzen. In the case n = 2, we completely solved the system in a certain space of power series.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

On a nonlinear volterra equation

Nonnegative solutions u of the nonlinear Volterra equation u = a * g(u) (g(0) = 0) in mathematical physics are considered. Under certain assumptions the nonhomogenuous equation u = a * g(u) + ? is studied. Some approximations of nonnegative solutions of the homogenuous equation are considered by the nonnegative solutions of the nonhomogenuous one.     

Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2 Distribution

Consider the test problem about matrix normal mean M with the null hypothesis M = O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379–1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of ² distributions referred to as X² distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the X² distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).     

Nonnegatively realizable spectra with two positive eigenvalues

Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n ? 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.     

Maximum principles for real (p, p)-forms on K?hler manifolds

In this paper, we extend the maximum principle for (1, 1)-Hermitian symmetric tensor to a complete K?hler manifold with bounded holomorphic bisectional curvature and nonnegative orthogonal bisectional curvature. We also achieve a maximum principle for real (p, p)-forms on a compact K?hler manifold with nonnegative holomorphic sectional curvature and vanishing Bochner tensor.     

König–Egerváry Graphs are Non-Edmonds

König–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors ${{\bf{x}}\in\mathbb R^E}K?nig–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors x ? \mathbb R^E{{\bf{x}}\in\mathbb R^E} satisfying two families of constraints: ‘vertex saturation’ and ‘blossom’. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs—one combinatorial, one algorithmic—of its title’s assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. (J Combin Theory Ser B 92:319–324, 2004).     

An application of prophet regions to optimal stopping with a random number of observations

Let X ₁,X ₂ ,?…?be any sequence of nonnegative integrable random variables, and let N∈{1,2 , …} be a random variable with known distribution, independent of X ₁,X ₂ , …. The optimal stopping value sup _t E(X_t I(N≥ t)) is considered for two players: one who has advance knowledge of the value of N, and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz [T.P. Hill and R.P. Kertz (1983). Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc., 278, 197–207].     

SUBGROUP DISTORTIONS IN NILPOTENT GROUPS

The explicit formula for the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group is obtained. In particular, we prove that a function f: N → R can be realized (up to equivalence) as the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group if and only if f ～ n^r for some nonnegative r ∈ Q. Considering lattices in Lie groups, we establish the analogous results for finitely generated nilpotent groups.     

Singularity and decay estimates for the conformal k-hessian equation via Liouville-type theorems

We study some connections between Liouville type theorems and local properties of nonnegative solutions to conformal k-hessian equations by making use of an elementary lemma for all positive functions in Li and Zhang (J. Anal. Math. 90 (2003), 27–87) and related Liouville type theorems in Li and Li (Acta. Math. 195 (2005), 117–154). Research of the second author is supported by Tianyuan Fund of Mathematics (10826061).     

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 ^s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 ^-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities. Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Complexity analysis and optimization of the shortest path tour problem

The shortest path tour problem (SPTP) consists in finding a shortest path from a given origination node s to a given destination node d in a directed graph with nonnegative arc lengths with the constraint that the optimal path P should successively and sequentially pass through at least one node from given node subsets T ₁, T ₂, . . . , T _N , where T_i ?T_j = ?, " i, j=1,?,N, i 1 j{T_i \cap T_j = \emptyset, \forall\ i, j=1,\ldots,N,\ i \neq j}. In this paper, it will proved that the SPTP belongs to the complexity class P and several alternative techniques will be presented to solve it.     

Pure ℓ-Ideals in Lattice-Ordered Rings#

An ?-ideal I of a commutative lattice-ordered ring R with positive identity element is called a pure ?-ideal if R  =  I  + ?( x ) for each x  ∈  I , where ?(x) is the ?-annihilator of x in R . In this article, we give some results on pure ?-ideals and study the ?-ideal structure of a commutative lattice-ordered ring with positive identity element by using pure ?-ideals.     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

A hybrid algorithm for solving linear inequalities in a least squares sense

The need for solving a system of linear inequalities, A x ≥ b, arises in many applications. Yet in some cases the system to be solved turns out to be inconsistent due to measurement errors in the data vector b. In such a case it is often desired to find the smallest correction of b that recovers feasibility. That is, we are looking for a small nonnegative vector, y ≥ 0, for which the modified system A x ≥ b - y is solvable. The problem of calculating the smallest correction vector is called the least deviation problem. In this paper we present new algorithms for solving this problem. Numerical experiments illustrate the usefulness of the proposed methods.     

On Fractional GJMS Operators

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet‐to‐Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli‐Silvestre extension for (?Δ)^γ when γ ? (0,1), and both a geometric interpretation and a curved analogue of the higher‐order extension found by R. Yang for (?Δ)^γ when γ > 1. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré‐Einstein manifold, including an interpretation as a renormalized energy. Second, for γ ? (1,2), we show that if the scalar curvature and the fractional Q‐curvature Q_2γ of the boundary are nonnegative, then the fractional GJMS operator P_2γ is nonnegative. Third, by assuming additionally that Q_2γ is not identically zero, we show that P_2γ satisfies a strong maximum principle.© 2016 Wiley Periodicals, Inc.