I-radicals and right perfect rings

We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of Komarnitskiy. We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings.Визначено кільця, для яких кожна теорія скруту з успадкуванням є теорією S-скруту у сенсі Комарницького. Показано, що такі кільця допускають первинний розклад. Комарницький отримав цей результат у частинному випадку лівих дуо-кілець

Categories of lattices, and their global structure in terms of almost split sequences

A major part of Iyama’s characterization of Auslander-Reiten quivers of representation-finite orders Λ consists of an induction via rejective subcategories of Λ-lattices, which amounts to a resolution of Λ as an isolated singularity. Despite of its useful applications (proof of Solomon’s second conjecture and the finiteness of representation dimension of any artinian algebra), rejective induction cannot be generalized to higher dimensional Cohen-Macaulay orders Λ. Our previous characterization of finite Auslander-Reiten quivers of Λ in terms of additive functions [22] was proved by means of L-functors, but we still had to rely on rejective induction. In the present article, this dependence will be eliminated

Arithmetic properties of exceptional lattice paths

For a fixed real number ρ > 0, let L be an affine line of slope ρ ⁻¹ in R ² . We show that the closest approximation of L by a path P in Z ² is unique, except in one case, up to integral translation. We study this exceptional case. For irrational ρ, the projection of P to L yields two quasicrystallographic tilings in the sense of Lunnon and Pleasants [5]. If ρ satisfies an equation x ² = mx + 1 with m ∈ Z, both quasicrystals are mapped to each other by a substitution rule. For rational ρ, we characterize the periodic parts of P by geometric and arithmetic properties, and exhibit a relationship to the hereditary algebras Hρ(K) over a field K introduced in a recent proof of a conjecture of Ro˘ıter

Lagrangian Reachabililty

We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an over-approximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.Comment: Accepted to CAV 201

Sharper and Simpler Nonlinear Interpolants for Program Verification

Interpolation of jointly infeasible predicates plays important roles in various program verification techniques such as invariant synthesis and CEGAR. Intrigued by the recent result by Dai et al.\ that combines real algebraic geometry and SDP optimization in synthesis of polynomial interpolants, the current paper contributes its enhancement that yields sharper and simpler interpolants. The enhancement is made possible by: theoretical observations in real algebraic geometry; and our continued fraction-based algorithm that rounds off (potentially erroneous) numerical solutions of SDP solvers. Experiment results support our tool's effectiveness; we also demonstrate the benefit of sharp and simple interpolants in program verification examples

Skew Left Braces of Nilpotent Type

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.Comment: 27 pages. Accepted for publication in Proc. London Math. Soc. (3

Some closure operations in Zariski-Riemann spaces of valuation domains: a survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains