1,161 research outputs found
Open -spin theory II: The analogue of Witten's conjecture for -spin disks
We conclude the construction of -spin theory in genus zero for Riemann
surfaces with boundary. In particular, we define open -spin intersection
numbers, and we prove that their generating function is closely related to the
wave function of the th Gelfand--Dickey integrable hierarchy. This provides
an analogue of Witten's -spin conjecture in the open setting and a first
step toward the construction of an open version of Fan--Jarvis--Ruan--Witten
theory. As an unexpected consequence, we establish a mysterious relationship
between open -spin theory and an extension of Witten's closed theory.Comment: The more foundational parts of the previous version, v3, were moved
to the article arXiv:2003.01082. These include the description of objects,
constructions of moduli spaces and bundles and proofs of orientations
theorems. The name of the paper and the abstract were changed accordingl
The combinatorial formula for open gravitational descendents
In recent works, [20],[21], descendent integrals on the moduli space of
Riemann surfaces with boundary were defined. It was conjectured in [20] that
the generating function of these integrals satisfies the open KdV equations. In
this paper we develop the notions of symmetric Strebel-Jenkins differentials
and of Kasteleyn orientations for graphs embedded in open surfaces. In addition
we write an explicit expression for the angular form of the sum of line
bundles. Using these tools we prove a formula for the descendent integrals in
terms of sums over weighted graphs. Based on this formula, the conjecture of
[20] was proved in [5]
Harmonic Labeling of Graphs
Which graphs admit an integer value harmonic function which is injective and
surjective onto ? Such a function, which we call harmonic labeling, is
constructed when the graph is the square grid. It is shown that for any
finite graph containing at least one edge, there is no harmonic labeling of
A Deep Hierarchical Approach to Lifelong Learning in Minecraft
We propose a lifelong learning system that has the ability to reuse and
transfer knowledge from one task to another while efficiently retaining the
previously learned knowledge-base. Knowledge is transferred by learning
reusable skills to solve tasks in Minecraft, a popular video game which is an
unsolved and high-dimensional lifelong learning problem. These reusable skills,
which we refer to as Deep Skill Networks, are then incorporated into our novel
Hierarchical Deep Reinforcement Learning Network (H-DRLN) architecture using
two techniques: (1) a deep skill array and (2) skill distillation, our novel
variation of policy distillation (Rusu et. al. 2015) for learning skills. Skill
distillation enables the HDRLN to efficiently retain knowledge and therefore
scale in lifelong learning, by accumulating knowledge and encapsulating
multiple reusable skills into a single distilled network. The H-DRLN exhibits
superior performance and lower learning sample complexity compared to the
regular Deep Q Network (Mnih et. al. 2015) in sub-domains of Minecraft
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