ARTICLE | doi:10.20944/preprints202303.0023.v1
Subject: Computer Science And Mathematics, Data Structures, Algorithms And Complexity Keywords: Game Design; Variational AutoEncoder (VAE); Image and Video Generation; Bayesian Algorithm; Loss Function; Data Clustering; Data and Image Analytics; MNIST database; Generator and Discriminator
Online: 1 March 2023 (11:17:12 CET)
In recent decades, the Variational AutoEncoder (VAE) model has shown good potential and capabilities in image generation and dimensionality reduction. The combination of VAE and various machine learning frameworks has also worked effectively in different daily life applications, however its possibility and effectiveness in modern game design has seldom been explored nor assessed. The use of its feature extractor for data clustering was minimally discussed in literature neither. This paper first attempts to explore different mathematical properties of the VAE model, in particular, the theoretical framework of the encoding and decoding processes, the possible achievable lower bound and loss functions of different applications; then applies the established VAE model into generating new game levels within two well-known game settings; as well as validating the effectiveness of its data clustering mechanism with the aid of the Modified National Institute of Standards and Technology (MNIST) database. Respective statistical metrics and assessments were also utilized for evaluating the performance of the proposed VAE model in aforementioned case studies. Based on the statistical and spatial results, several potential drawbacks and future enhancement of the established model were outlined, with the aim of maximizing the strengths and advantages of VAE for future game design tasks and relevant industrial missions.
ARTICLE | doi:10.20944/preprints201810.0213.v1
Subject: Environmental And Earth Sciences, Atmospheric Science And Meteorology Keywords: Remote Sensing Techniques; Tropospheric NO2 Column Retrieval; Air Mass Factor (AMF); Meteorological Reformulation; MAX-DOAS measurements; Satellite Informatics
Online: 10 October 2018 (10:08:58 CEST)
Improving air quality and reducing human exposure to unhealthy levels of airborne chemicals are important global missions, particularly in China. Satellite remote sensing offers a powerful tool to examine regional trends in NO2, thus providing a direct measure of key parameters that strongly affect surface air quality. To accurately resolve spatial gradients in NO2 concentration using satellite observations and thus understand local and regional aspects of air quality, a priori input data at sufficiently high spatial and temporal resolution to account for pixel-to-pixel variability in the characteristics of the land and atmosphere are required. In this paper, we adapt the Berkeley High Resolution product (BEHR v3.0A, v3.0B and v3.0C) and meteorological outputs from the Weather Research and Forecasting (WRF) model to describe column NO2 in southern China. The BEHR approach is particularly useful for places with large spatial variabilities and terrain height differences such as China. We retrieved tropospheric NO2 vertical column density (TVCD) within part of southern China, for four seasons of 2015, based upon satellite datasets from Ozone Monitoring Instrument (OMI). Retrieval results are validated by comparing with MAX-DOAS tropospheric column measurements conducted in Guangzhou. BEHR retrieval algorithms are more consistent with MAX-DOAS measurements than OMI-NASA retrieval, opening new windows into research questions that require high spatial resolution, for example retrieving NO2 vertical column and ground pollutant concentration in China and other countries.
ARTICLE | doi:10.20944/preprints202302.0522.v1
Subject: Computer Science And Mathematics, Discrete Mathematics And Combinatorics Keywords: fractional-order stochastic dominance; discrete stochastic dominance; discrete utility; fractional sum
Online: 28 February 2023 (14:22:40 CET)
A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In early days, Fishburn investigated SD from the view of distributions and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD may not be generalized directly since some properties of fractional calculus do not have a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be generalized directly from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle them.
ARTICLE | doi:10.20944/preprints202206.0331.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Alien Tiles; Coding Theory; Lights Out; Recreational Mathematics; Abstract Algebra
Online: 24 June 2022 (05:13:05 CEST)
The switching game Lights Out and its variants were studied extensively as recreational mathematics problems. The game board of the ordinary Lights Out is a rectangular grid of lights, where each light is either on or off. By clicking a light, the clicked light and its adjacent rectilinear neighbors are toggled. Given an arbitrary initial configuration of lights, the final mission is to “solve” this game by switching off all the lights. Most studies on Lights Out and its variants focused on the solvability of given games or the number of solvable games, but when the game is viewed in a coding-theoretical perspective, more interesting questions with special meanings in coding theory will naturally pop up, such as finding the minimal number of lit lights among all solvable games except the solved game, or finding the minimal number of lit lights that the player can achieve from a given unsolvable game, etc. However, these problems are usually hard to be solved in general in terms of algorithmic complexity. This study considers a natural extension of the Lights Out game, which enlarges the toggle pattern in a way that all the lights in the same row and those in the same column of the clicked light are toggled. This variant of Lights Out is a two-state version of a switching game called Alien Tiles. In this paper, we investigate the properties of the two-state Alien Tiles, and discuss several coding-theoretical problems about this game. Then, we apply this game as an error correction code and investigate its optimality. We also give a brief overview on algorithmic complexity and coding theory for readers who are not familiar with these topics. The purpose of this paper is to propose ways of playing switching games in a think-outside-the-box manner, which benefit the recreational mathematics community.