Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets

Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where non-convex curves are necessary, our method draws an appropriate diagram using non-convex polygons. Thus, we are now always able to automatically visualize data for up to three sets

Investigating diagrammatic reasoning with deep neural networks

Diagrams in mechanised reasoning systems are typically en- coded into symbolic representations that can be easily processed with rule-based expert systems. This relies on human experts to define the framework of diagram-to-symbol mapping and the set of rules to reason with the symbols. We present a new method of using Deep artificial Neu- ral Networks (DNN) to learn continuous, vector-form representations of diagrams without any human input, and entirely from datasets of dia- grammatic reasoning problems. Based on this DNN, we developed a novel reasoning system, Euler-Net, to solve syllogisms with Euler diagrams. Euler-Net takes two Euler diagrams representing the premises in a syl- logism as input, and outputs either a categorical (subset, intersection or disjoint) or diagrammatic conclusion (generating an Euler diagram rep- resenting the conclusion) to the syllogism. Euler-Net can achieve 99.5% accuracy for generating syllogism conclusion. We analyse the learned representations of the diagrams, and show that meaningful information can be extracted from such neural representations. We propose that our framework can be applied to other types of diagrams, especially the ones we don’t know how to formalise symbolically. Furthermore, we propose to investigate the relation between our artificial DNN and human neural circuitry when performing diagrammatic reasoning

Partial chord diagrams and matrix models

In this article, the enumeration of partial chord diagrams is discussed via matrix model techniques. In addition to the basic data such as the number of backbones and chords, we also consider the Euler characteristic, the backbone spectrum, the boundary point spectrum, and the boundary length spectrum. Furthermore, we consider the boundary length and point spectrum that unifies the last two types of spectra. We introduce matrix models that encode generating functions of partial chord diagrams filtered by each of these spectra. Using these matrix models, we derive partial differential equations - obtained independently by cut-and-join arguments in an earlier work - for the corresponding generating functions.Comment: 42 pages, 14 figure

Generating Effective Euler Diagrams

Euler diagrams are used for visualizing categorized data,with applications including crime control, bioinformatics, classification systems and education. Various properties of Euler diagrams have been empirically shown to aid, or hinder, their comprehension by users. Therefore, a key goal is to automatically generate Euler diagrams that possess beneficial layout features whilst avoiding those that are a hindrance.The automated layout techniques that currently exist sometimes produce diagrams with undesirable features. In this paper we present a novel approach, called iCurves, for generating Euler diagrams alongside a prototype implementation. We evaluate iCurves against existing techniques based on the aforementioned layout properties. This evaluation suggests that, particularly when the number of zones is high, iCurves can outperform other automated techniques in terms of effectiveness for users, as indicated by the layout properties of the produced Euler diagrams

Enumeration of diagonally colored Young diagrams

In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews' work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.Comment: Final version, 12 pages. To appear in Monatshefte f\"ur Mathemati

On a class of polynomial Lagrangians

In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincar\'e--Cartan and Euler--Lagrange forms, momenta and momenta of generating forms, a new geometric object arising in variational sequences. Finally, we provide a simple but important example of special Lagrangian, namely the Hilbert--Einstein Lagrangian.Comment: LaTeX2e, amsmath, diagrams, hyperref; 15 page