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In this paper, an explanation of the Newton-Peiseux algorithm is given. This explanation is supplemented with well-worked and explained examples of how to use the algorithm to find fractional power series expansions for all branches of a polynomial at the origin.
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Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a more subtle result than is often appreciated, since it applies to computations run on qubit-based quantum computers in the single walker case, and physical quantum walks in the multi-walker case (quantum cellular automata). Nonetheless, quantum walks are powerful tools for quantum computing when correctly applied. In this paper, I explain the relationship between quantum walks as models and quantum walks as computational tools, and give some examples of their application in both contexts.
Synchronization is known to play a vital role within many highly connected neural systems such as the olfactory systems of fish and insects. In this paper we show how one can robustly and effectively perform practical computations using small perturbations to a very simple globally coupled network of coupled oscillators. Computations are performed by exploiting the spatio-temporal dynamics of a robust attracting heteroclinic network (also referred to as `winnerless competition dynamics). We use different cluster synchronization states to encode memory states and use this to design a simple multi-base counter. The simulations indicate that this gives a robust computational system exploiting the natural dynamics of the system.
We present our implementation of an algorithm which functions as a numerical oracle for the Newton polytope of a hypersurface in the Macaulay2 package NumericalNP.m2. We propose a tropical membership test, relying on this algorithm, for higher codimension varieties based on ideas from Hept and Theobald. To showcase this software, we investigate the Newton polytope of both a hypersurface coming from algebraic vision and the Luroth invariant.
PointGoal navigation has seen significant recent interest and progress, spurred on by the Habitat platform and associated challenge. In this paper, we study PointGoal navigation under both a sample budget (75 million frames) and a compute budget (1 GPU for 1 day). We conduct an extensive set of experiments, cumulatively totaling over 50,000 GPU-hours, that let us identify and discuss a number of ostensibly minor but significant design choices -- the advantage estimation procedure (a key component in training), visual encoder architecture, and a seemingly minor hyper-parameter change. Overall, these design choices to lead considerable and consistent improvements over the baselines present in Savva et al. Under a sample budget, performance for RGB-D agents improves 8 SPL on Gibson (14% relative improvement) and 20 SPL on Matterport3D (38% relative improvement). Under a compute budget, performance for RGB-D agents improves by 19 SPL on Gibson (32% relative improvement) and 35 SPL on Matterport3D (220% relative improvement). We hope our findings and recommendations will make serve to make the communitys experiments more efficient.
Let $k$ be either a number a field or a function field over $mathbb{Q}$ with finitely many variables. We present a practical algorithm to compute the geometric Picard lattice of a K3 surface over $k$ of degree $2$, i.e., a double cover of the projective plane over $k$ ramified above a smooth sextic curve. The algorithm might not terminate, but if it terminates then it returns a proven correct answer.
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