An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure

What Twitter Profile and Posted Images Reveal About Depression and Anxiety

Previous work has found strong links between the choice of social media images and users' emotions, demographics and personality traits. In this study, we examine which attributes of profile and posted images are associated with depression and anxiety of Twitter users. We used a sample of 28,749 Facebook users to build a language prediction model of survey-reported depression and anxiety, and validated it on Twitter on a sample of 887 users who had taken anxiety and depression surveys. We then applied it to a different set of 4,132 Twitter users to impute language-based depression and anxiety labels, and extracted interpretable features of posted and profile pictures to uncover the associations with users' depression and anxiety, controlling for demographics. For depression, we find that profile pictures suppress positive emotions rather than display more negative emotions, likely because of social media self-presentation biases. They also tend to show the single face of the user (rather than show her in groups of friends), marking increased focus on the self, emblematic for depression. Posted images are dominated by grayscale and low aesthetic cohesion across a variety of image features. Profile images of anxious users are similarly marked by grayscale and low aesthetic cohesion, but less so than those of depressed users. Finally, we show that image features can be used to predict depression and anxiety, and that multitask learning that includes a joint modeling of demographics improves prediction performance. Overall, we find that the image attributes that mark depression and anxiety offer a rich lens into these conditions largely congruent with the psychological literature, and that images on Twitter allow inferences about the mental health status of users.Comment: ICWSM 201

An Auction-Based Method for Decentralized Train Scheduling

We present a computational study of an auction-based method for decentralized train scheduling. The method is well suited to the natural information and control structure of mod- ern railroads. We assume separate network territories, with an autonomous dispatch agent responsible for the ow of trains over each territory. Each train is represented by a self-interested agent that bids for the right to travel across the network from its source to destination, submitting bids to multiple dispatch agents along its route as necessary. The bidding language allows trains to bid for the right to enter and exit territories at particular times, and also to represent indifference over a range of times. Computational results on a simple network with straight-forward best-response bid- ding strategies demonstrate that the auction computes near- optimal system-wide schedules. In addition, the method appears to have useful scaling properties, both with the number of trains and with the number of dispatchers, and generates less extremal solutions than those obtained using traditional centralized optimization techniques.Engineering and Applied Science

Growth and Survival of the Halophyte Salicornia Europaera L. Under Saline Field Conditions

Author Institution: Department of Botany, Ohio UniversityField investigations were carried out to determine growth and survival rates of Salicornia europaea L. in a saline environment at Rittman, Ohio. Collected data indicated that from 62% to 100% of the seedlings within the 'A saline zones investigated did not survive to maturity. Seedling mortality was statistically correlated at P<0.01 to rising soil salinity stress during late spring and summer. Plant growth was minimal between April and June, increasing sharply during late summer and fall

An Ascending-Price Generalized Vickrey Auction

A simple characterization of the equilibrium conditions required to compute Vickrey payments in the Combinatorial Allocation Problem leads to an ascending price Generalized Vickrey Auction. The ascending auc- tion, iBundle Extend & Adjust (iBEA), maintains non-linear and perhaps non-anonymous prices on bundles of items, and terminates with the ef- cient allocation and the Vickrey payments in ex post Nash equilibrium. Crucially, iBEA is able to implement the Vickrey outcome even when the Vickrey payments are not supported in a single competitive equilibrium. The auction closes with Universal competitive equilibrium prices, which provide enough information to compute individualized discounts to adjust the nal prices and implement Vickrey payments.Engineering and Applied Science

Preventing Strategic Manipulation in Iterative Auctions: Proxy Agents and Price-Adjustment

Iterative auctions have many computational advantages over sealed-bid auctions, but can present new possibilities for strategic manipulation. We propose a two-stage technique to make iterative auctions that compute optimal allocations with myopic best-response bidding strategies more robust to manipulation. First, introduce proxy bidding agents to constrain bidding strategies to (possibly untruthful) myopic bestresponse. Second, after the auction terminates adjust the prices towards those given in the Vickrey auction, a sealedbid auction in which truth-revelation is optimal. We present an application of this methodology to iBundle, an iterative combinatorial auction which gives optimal allocations for myopic best-response agents.Engineering and Applied Science

Iterative Combinatorial Auctions: Theory and Practice

Combinatorial auctions, which allow agents to bid directly for bundles of resources, are necessary for optimal auction-based solutions to resource allocation problems with agents that have non-additive values for resources, such as distributed scheduling and task assignment problems. We introduce iBundle, the first iterative combinatorial auction that is optimal for a reasonable agent bidding strategy, in this case myopic best-response bidding. Its optimality is proved with a novel connection to primal-dual optimization theory. We demonstrate orders of magnitude performance improvements over the only other known optimal combinatorial auction, the Generalized Vickrey Auction.Engineering and Applied Science

Learning and Adaption in Multiagent Systems

The goal of a self-interested agent within a multiagent system is to maximize its utility over time. In a situation of strategic interdependence, where the actions of one agent may a ect the utilities of other agents, the optimal behavior of an agent must be conditioned on the expected behaviors of the other agents in the system. Standard game theory assumes that the rationality and preferences of all the agents is common knowledge: each agent is then able to compute the set of possible equilibria, and if there is a unique equilibrium, choose a best-response to the actions that the other agents will all play. Real agents acting within a multiagent system face multiple problems: the agents may have incomplete information about the preferences and rationality of the other agents in the game, computing the equilibria can be computationally complex, and there might be many equilibria from which to choose. An alternative explanation of the emergence of a stable equilibrium is that it arises as the long-run outcome of a repeated game, in which bounded-rational agents adapt their strategies as they learn about the other agents in the system. We review some possible models of learning for games, and then show the pros and cons of using learning in a particular game, the Compensation Mechanism, a mechanism for the efficient coordination of actions within a multiagent system.Engineering and Applied Science

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$