Related papers: The period-index problem for elliptic curves and t…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…

It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has…

We consider a particular case of an analog for elliptic curves to the Mersenne problem : finding explicitely all prime power terms in an elliptic divisibility sequence when descent via isogeny is possible. We explain how this question can…

We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear…

In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some…

A standard scientific study comprises two processes: one is to describe a thing, and the other is to understand how the thing works. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe…

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few…

We study four fundamental questions about $1$-periods and give complete answers. 1) We give a necessary and sufficient for a period integral to be transcendental. 2) We give a qualitative description of all $\overline{\mathbf{Q}}$-linear…

In this paper, we investigate semilinear elliptic equations with general exponential-type nonlinearities in two dimensions. For such nonlinearities, we establish two main results. The first is the construction of a singular solution.…

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing…

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian…

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated…

In this work we describe and compare the classic inner product and Pearson correlation coefficient as well as the recently introduced real-valued Jaccard and coincidence indices. Special attention is given to diverse schemes for taking into…

We use twisted relative Picard varieties to split Brauer classes on projective varieties over algebraically closed fields by torsors for a fixed abelian scheme independent of the Brauer class. The construction is also used to prove that the…

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit…

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze…

We prove a generic Torelli theorem for Jacobian elliptic surfaces, provided that the geometric genus is large compared to the irregularity. The result is effective to the extent that defining equations for the base curve are recovered from…

The purpose of this short note is to establish the existence of $\partial$-parameterized Picard-Vessiot extensions of systems of linear difference-differential equations over difference-differential fields with algebraically closed…