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We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even line-transitivity for the horocyclic flow. We also describe…
We give an explicit construction of the p-adic de Rham comparison isomorphism for 1-motives. In particular, we prove that our construction recovers the classical de Rham comparison isomorphism and is functorial with respect to morphisms of…
Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of…
We prove that the projective complex algebraic varieties admitting a large complex local system satisfy a strong version of the Green-Griffiths-Lang conjecture.
In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable…
We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are…
We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the…
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular…
We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic…
We consider a class of discontinuous piecewise linear differential systems in $\mathbb{R}^3$ with two pieces separated by a plane. In this class we show that there exist differential systems having: a unique limit cycle, a unique…
Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li…
We establish several delay-independent criteria for the existence and stability of positive periodic solutions of n-dimensional nonautonomous functional differential equation by several fixed point theorems. Examples from positive and…
The period set of a dynamical system is defined as the subset of all integers $n$ such that the system has a periodic orbit of length $n$. Based on known results on the intersection of period sets of torus maps within a homotopy class, we…
We classify rational cuspidal curves of degrees 6 and 7 in the complex projective plane, up to symplectic isotopy. The proof uses topological tools, pseudoholomorphic techniques, and birational transformations.
We introduce and study the notion of universally defined cycles of smooth varieties of dimension $d$, and prove that they are given by polynomials in the Chern classes. A similar result is proved for universally defined cycles on products…
We investigate global continuation of periodic orbits of a differential equation depending on a parameter, assuming that a closed 1-form satisfying certain properties exists. We begin by extending the global continuation theory of…
We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial…
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
By studying periodic points for rational maps on $\bm{C}^d$ with $p$ invariants, we show that they form an invariant variety of dimension $p$ if the periodicity conditions are `fully correlated', and a set of isolated points if the…
We present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary.