Iwasawa theory for elliptic curves at supersingular primes

We give a new formulation in Iwasawa theory for elliptic curves at good supersingular primes. This formulation is similar to Mazur’s at good ordinary primes. Namely, we define a new Selmer group, and show that it is of Λ-cotorsion. Then we formulate the Iwasawa main conjecture as that the characteristic ideal is generated by Pollack’s p-adic L-function. We show that this main conjecture is equivalent to Kato’s and Perrin-Riou’s main conjectures. We also prove an inequality in the main conjecture by using Kato’s Euler system. In terms of the λ- and the μ-invariants of our Selmer group, we specify the numbers λ and μ in the asymptotic formula for the order of the Tate-Shafarevich group by Kurihara and Perrin-Riou. Oblatum 17-VI-2002 & 2-IX-2002?Published online: 18 December 2002     

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.     

Maximum Principles for Infinite Dimensional Diffusion Equations

In this paper we prove the validity of the Maximum Principle for some class of elliptic and parabolic equations of diffusion type in infinite dimension. The main tools are Asplund’s theorem and Preiss’ theorem on differentiability of Lipschitz functions in Banach space.      

On singular nonhomogeneous elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent

In this paper, we establish the existence of multiple solutions for nonhogeneous singular elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent, by using Ekeland’s Variational Principle and Mountain Pass Theorem without Palais Smale conditions.     

Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n , II

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.     

Ranks of twists of elliptic curves and Hilbert’s tenth problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.     

Multiplicity results of p(x)-Laplacian systems with Neumann conditions

In this paper, we study a Neumann problem for elliptic systems with variable exponents. We obtain the existence of at least three nontrivial solutions by using an equivalent variational approach to a recent Ricceri’s three critical points theorem (Ricceri in Nonlinear Anal TMA 70:3084–3089, 2009).     

The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

A variational inequality with mixed boundary conditions

In this paper we study a variational inequality for a second order uniformly elliptic operator on a bounded domain, the solution of which is required to lie above a given obstacle and to assume assigned values on a part of the boundary of the domain. We are mainly concerned with the regularity of the solution in relation to the regularity of the data. During the preparation of the paper the authors were partially supported by the Italian Consiglio Nazionale delle Ricerche, the first as visiting professor at the Scuola Normale Superiore (Pisa) on deputation from the Tata Institute of Fundamental Research (Bombay) and the second through the Istituto per l’Elaborazione dell’Informazione (Pisa).     

Integral representation of the solution of a Gellerstedt problem

The solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation was earlier obtained in the form of a series for the case of a half-strip in the elliptic part of the domain and of nonzero boundary data on the characteristics in the hyperbolic part of the domain. In the present paper, we give an integral representation of the solutions of this problem as well as some related problems.     

Embedding Distributions in Algebras of Generalized Functions with Singularities

 It is a known open question whether in Rosinger’s nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger’s algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions. Further, we indicate that most of the properties of Rosinger’s algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties.     

Embedding Distributions in Algebras of Generalized Functions with Singularities

 It is a known open question whether in Rosinger’s nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger’s algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions. Further, we indicate that most of the properties of Rosinger’s algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties. Received September 20, 2001; in final form April 11, 2002     

Counting the order of the group of points of an elliptic curve over a finite field based on Shanks’s algorithm

Shanks’s baby-step-giant-step algorithm to count the order of the group of points of an elliptic curve over a finite prime field is described. The algorithm’s improvements are also discussed. The first one is based on Mestre’s theorem. The second improvement is based on the Sutherland’s probabilistic result allowing to calculate the exponent of an arbitrary finite abelian group.     

On the proof of the reciprocity law for arithmetic Siegel modular functions

Earlier we obtained a new proof of Shimura’s reciprocity law for the special values of arithmetic Hilbert modular functions. In this note we show how from this result one may derive Shimura’s reciprocity law for special values of arithmetic Siegel modular functions. To achieve this we use Shimura’s classification of the special points of the Siegel space, Satake’s classification of the equivariant holomorphic imbeddings of Hilbert-Siegel modular spaces into a larger Siegel space, and, finally, a corrected version of some of Karel’s results giving an action of the Galois group Gal(Q_ab/Q) on arithmetic Siegel modular forms. Research supported in part by the NSF Grant No. DMS-8601130.     

An Uncertainty Principle for Eigenfunction Expansions

In this paper, we extend a theorem of Hardy’s on Fourier transform pairs to: (a) a noncompact-type Riemannian symmetric space of rank one, with respect to the eigenfunction expansion of the invariant Laplacian; (b) a compact Riemannian manifold with respect to the eigenfunction expansion of a positive elliptic operator; and (c) Rⁿ with respect to Hermite and Laguerre expansions.     

Stability of join-the-shortest-queue networks

This paper investigates stability behavior in a variant of a generalized Jackson queueing network. In our network, some customers use a join-the-shortest-queue policy when entering the network or moving to the next station. Furthermore, we allow interarrival and service times to have general distributions. For networks with two stations we derive necessary and sufficient conditions for positive Harris recurrence of the network process. These conditions involve only the mean values of the network primitives. We also provide counterexamples showing that more information on distributions and tie-breaking probabilities is needed for networks with more than two stations, in order to characterize the stability of such systems. However, if the routing probabilities in the network satisfy a certain homogeneity condition, then we show that the stability behavior can be explicitly determined, again using the mean value parameters of the network. A byproduct of our analysis is a new method for using the fluid model of a queueing network to show non-positive recurrence of a process. In previous work, the fluid model was only used to show either positive Harris recurrence or transience of a network process. J.G. Dai’s research supported in part by National Science Foundation grants DMI-0300599, CMMI-0727400 and CNS-0718701, and by an IBM Faculty Award. J.J. Hasenbein’s research supported in part by National Science Foundation grant DMI-0132038. B. Kim’s research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

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     

Ratio of Generalized Hill’s estimator and its asymptotic normality theory

We present a statistical process depending on a continuous time parameter τ whose each margin provides a Generalized Hill’s estimator. In this paper, the asymptotic normality of the finite-dimensional distributions of this family are completely characterized for τ > 1/2 when the underlying distribution function lies on the maximum domain of attraction. The ratio of two different margins of the statistical process characterizes entirely the whole domain of attraction. Its asymptotic normality is also studied. The results permit in general to build a new family of estimators for the extreme value index whose asymptotic properties can be easily derived. For example, we give a new estimate of the Weibull extreme value index and we study its consistency and its asymptotic normality.