On modular solutions of certain meromorphic modular differential equations

Kaneko and Koike gave the “extremal” quasimodular forms of depth 1 for PSL₂(ℤ) and modular differential equations they satisfy. In this paper, we study modular solutions of their modular differential equations.     

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.     

Distributions and analytic continuation of Dirichlet series

This is the second in a series of three papers; the other two are “Summation Formulas, from Poisson and Voronoi to the Present” [Progr. Math. 220 (2004) 419-440] and “Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)” (preprint). The first paper is primarily expository, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on . The present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)×GL(1) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function, degree 1 and degree 2 L-functions, the converse theorem for GL(2), and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities.     

On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD

We study the initial-boundary value problem resulting from the linearization of the plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). We suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously, but jumps. For the basic state upon which we perform linearization we find two cases of well-posedness of the “frozen” coefficient problem: the “gas dynamical” case and the “purely MHD” case. In the “gas dynamical” case we assume that the jump of the normal derivative of the total pressure is always negative. In the “purely MHD” case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non-zero and non-parallel to each other everywhere on the interface. For this case we prove a basic a priori estimate in the anisotropic weighted Sobolev space for the variable coefficient problem.     

Dissipative Elliptic Problems in Domains with Cylindrical Ends, Scattering Matrices, and Radiation Conditions

In a domain with cylindrical ends at infinity, we consider a general elliptic dissipative boundary value problem. The coefficients of the imaginary part of the operator of the problem vanish as The asymptotic behavior of the solutions is expressed in terms of incoming and outgoing waves (the amplitudes of such waves can grow at infinity). We introduce an (augmented) scattering matrix and, in terms of this matrix, we compute the number of linearly independent solutions to the homogeneous problem vanishing at infinity with a given rate. We discuss the statement of a problem with the so-called radiation conditions. The natural radiation conditions (only outgoing waves occur in asymptotic formulas for solutions) can be applied in any case. Other admissible radiation conditions for the problem under consideration are connected with the natural ones via scattering matrices. Bibliography: 12 titles.     

Quantum statistical mechanics over function fields

In this paper we construct a noncommutative space of “pointed Drinfeld modules” that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld modules to possibly degenerate level structures. In the second part of the paper we develop some notions of quantum statistical mechanics in positive characteristic and we show that, in the case of Drinfeld modules of rank one, there is a natural time evolution on the associated noncommutative space, which is closely related to the positive characteristic L-functions introduced by Goss. The points of the usual moduli space of Drinfeld modules define KMS functionals for this time evolution. We also show that the scaling action on the dual system is induced by a Frobenius action, up to a Wick rotation to imaginary time.     

Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P.M. Gruber proposed the Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” In this paper we solve the generalized Ulam stability problem for non-linear Euler-Lagrange quadratic mappings satisfying approximately a mean equation and an Euler-Lagrange type functional equations in quasi-Banach spaces and p-Banach spaces.     

Some remarks on the geometry of austere manifolds

We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called austere and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition on the second fundamental form and essentially requires that the non-zero eigenvalues of the second fundamental form in any normal direction at any point occur in oppositely signed pairs. We solve the pointwise problem of describing the set of austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions.Partially supported by NSF grants DMS-8352009 and DMS-8905207.     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

On Laguerre form and Laguerre isoparametric hypersurfaces

Let x : Mn-1→ Rnbe an umbilical free hypersurface with non-zero principal curvatures.M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi–Civita connection of its Laguerre metric.     

Fundamental results for pointfree convex geometry

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.     

On the Ulam stability of Jensen and Jensen type mappings on restricted domains

In 1941 Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 Bourgin was the second author to treat this problem for additive mappings. In 1982-1998 Rassias established the Hyers-Ulam stability of linear and nonlinear mappings. In 1983 Skof was the first author to solve the same problem on a restricted domain. In 1998 Jung investigated the Hyers-Ulam stability of more general mappings on restricted domains. In this paper we introduce additive mappings of two forms: of “Jensen” and “Jensen type,” and achieve the Ulam stability of these mappings on restricted domains. Finally, we apply our results to the asymptotic behavior of the functional equations of these types.     

The height and range of watermelons without wall

We determine the weak limit of the distribution of the random variables “height” and “range” on the set of p$p$-watermelons without wall restriction as the number of steps tends to infinity. Additionally, we provide asymptotics for the moments of the random variable “height”.     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.     

On zeros of characteristic p zeta function

The location and multiplicity of the zeros of zeta functions encode interesting arithmetic information. We study the characteristic p zeta function of Goss. We focus on “trivial” zeros and prove a theorem on zeros at negative integers, showing more vanishing than that suggested by naive analogies. We also compute some concrete examples providing the extra vanishing, when the class number is more than one.Finally, we give an application of these results to the non-vanishing of certain class group components for cyclotomic function fields. In particular, we give examples of function fields, where all the primes of degree more than two are “irregular”, in the sense of the Drinfeld-Hayes cyclotomic theory.     

Generalized coalitional semivalues

In this paper we change some axioms in the axiom system which defines coalitional semivalues [Albizuri, M.J., Zarzuelo J.M., 2004. On coalitional semivalues. Games and Economic Behavior 49, 221–243] and we define generalized coalitional semivalues. Generalized coalitional semivalues, like coalitional semivalues, are “compositions” of semivalues, but they form a broader set of “compositions”. Like coalitional semivalues, generalized coalitional semivalues are extensions to the coalitional context of semivalues for transferable utility games [Dubey, P., Neyman, A., Weber, R.J., 1981. Value theory without efficiency. Mathematics of Operations Research 6, 122–128].     

Numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity

This paper studies the degenerate scale problem arising from multiple rigid lines in plane elasticity. In the first step, the problem should be formulated on a degenerate scale by distribution of body force densities along rigid lines. The condition of vanishing displacement along lines is also assumed. The coordinate transform with a reduced factor “h” is performed in the next step. The new obtained BIE is a particular non-homogenous BIE defined in the transformed coordinates with normal scale. In the normal scale, the integral operator is invertible. By using two fundamental solutions that are formulated in the normal scale, the new obtained BIE can be reduced to an equation for finding the factor “h”. Finally, the degenerate scale is obtained. It is proved from computed results that the degenerate scale only depends on the configuration of rigid lines, and does not depend on the initial normal scale used. In addition, the degenerate scale is invariant with respect to the rotation of rigid lines. Many examples are carried out.     

The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.