Asymptotic invariants of base loci

The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.Comment: 26 pages, 1 figure; shorter version with more efficient exposition and more general version of asymptotic invariants; notation changed, references added, minor mistakes correcte

Основи інформатики, інформаційні технології та комп'ютерна ергономіка для гуманітарних напрямків

Розглянуто загальні питання про основи інформатики, інформаційних технологій та комп'ютерної ергономіки. Висвітлено основні положення теоретичного матеріалу щодо роботи користувача в операційному середовищі персонального комп'ютера та ергономічної безпеки використання технічних пристроїв. Призначено для студентів спеціальностей 053 "Психологія", 011 "Освітні педагогічні науки", 074 "Публічне управління та адміністрування" денної та заочної форм навчання

Weak Riemannian manifolds from finite index subfactors

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $${\cal O}(p) =\{u p u^*: u\in U_M\}$$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p)$ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.Comment: 19 page

Hybrid actor-critic algorithm for quantum reinforcement learning at CERN beam lines

Free energy-based reinforcement learning (FERL) with clamped quantum Boltzmann machines (QBM) was shown to significantly improve the learning efficiency compared to classical Q-learning with the restriction, however, to discrete state-action space environments. In this paper, the FERL approach is extended to multi-dimensional continuous state-action space environments to open the doors for a broader range of real-world applications. First, free energy-based Q-learning is studied for discrete action spaces, but continuous state spaces and the impact of experience replay on sample efficiency is assessed. In a second step, a hybrid actor-critic scheme for continuous state-action spaces is developed based on the Deep Deterministic Policy Gradient algorithm combining a classical actor network with a QBM-based critic. The results obtained with quantum annealing, both simulated and with D-Wave quantum annealing hardware, are discussed, and the performance is compared to classical reinforcement learning methods. The environments used throughout represent existing particle accelerator beam lines at the European Organisation for Nuclear Research (CERN). Among others, the hybrid actor-critic agent is evaluated on the actual electron beam line of the Advanced Plasma Wakefield Experiment (AWAKE).Comment: 17 pages, 15 figures, to be submitted to "Quantum" journa