Monomial Complete Intersections, The Weak Lefschetz Property and Plane Partitions

We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in characteristic p. We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing M(a,b,c).Comment: 16 pages. Minor revisions, mainly to keep track of two interesting developments following the original posting. Final version to appear in Discrete Mat

Efficiency and power of minimally nonlinear irreversible heat engines with broken time-reversal symmetry

We study the minimally nonlinear irreversible heat engines in which the time-reversal symmetry for the systems may b e broken. The expressions for the power and the efficiency are derived, in which the effects of the nonlinear terms due to dissipations are included. We show that, as within the linear responses, the minimally nonlinear irreversible heat engines enable attainment of Carnot efficiency at positive power. We also find that the Curzon-Ahlborn limit imposed on the efficiency at maximum power can be overcomed if the time-reversal symmetry is broken

Read, Watch, and Move: Reinforcement Learning for Temporally Grounding Natural Language Descriptions in Videos

The task of video grounding, which temporally localizes a natural language description in a video, plays an important role in understanding videos. Existing studies have adopted strategies of sliding window over the entire video or exhaustively ranking all possible clip-sentence pairs in a pre-segmented video, which inevitably suffer from exhaustively enumerated candidates. To alleviate this problem, we formulate this task as a problem of sequential decision making by learning an agent which regulates the temporal grounding boundaries progressively based on its policy. Specifically, we propose a reinforcement learning based framework improved by multi-task learning and it shows steady performance gains by considering additional supervised boundary information during training. Our proposed framework achieves state-of-the-art performance on ActivityNet'18 DenseCaption dataset and Charades-STA dataset while observing only 10 or less clips per video.Comment: AAAI 201