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Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
In this paper we propose two variants of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the new preconditioners, we use the simplest coarse solver…
Geometry problem solving presents distinctive challenges in artificial intelligence, requiring exceptional multimodal comprehension and rigorous mathematical reasoning capabilities. Existing approaches typically fall into two categories:…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
Multigrid solvers are the standard in modern scientific computing simulations. Domain Decomposition Aggregation-Based Algebraic Multigrid, also known as the DD-$\alpha$AMG solver, is a successful realization of an algebraic multigrid solver…
GeoGebra is an open source mathematics education software tool being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians…
Vectors of Locally Aggregated Descriptors (VLAD) have emerged as powerful image/video representations that compete with or even outperform state-of-the-art approaches on many challenging visual recognition tasks. In this paper, we address…
This paper presents an intelligent tutoring system, GeoTutor, for Euclidean Geometry that is automatically able to synthesize proof problems and their respective solutions given a geometric figure together with a set of properties true of…
Cylindrical Algebraic Decomposition (CAD) is an important tool within computational real algebraic geometry, capable of solving many problems to do with polynomial systems over the reals, but known to have worst-case computational…
Elliptic partial differential equations must be solved numerically for many problems in numerical relativity, such as initial data for every simulation of merging black holes and neutron stars. Existing elliptic solvers can take multiple…
Graph Neural Networks have recently become a prevailing paradigm for various high-impact graph analytical problems. Existing efforts can be mainly categorized as spectral-based and spatial-based methods. The major challenge for the former…
Recent machine reading comprehension datasets such as ReClor and LogiQA require performing logical reasoning over text. Conventional neural models are insufficient for logical reasoning, while symbolic reasoners cannot directly apply to…
This paper introduces an open-source software for distributed and decentralized non-convex optimization named ALADIN-$\alpha$. ALADIN-$\alpha$ is a MATLAB implementation of tailored variants of the Augmented Lagrangian Alternating Direction…
Nonlinear dimension reduction methods provide a low-dimensional representation of high-dimensional data by applying a Nonlinear transformation. However, the complexity of the transformations and data structures can create wildly different…
Kingdon is an open-source Python package designed to seamlessly integrate Geometric Algebra (GA) into existing workflows. Unlike previous GA libraries, kingdon is input-type-agnostic, and hence supports GA's over e.g. PyTorch tensors, NumPy…
Self-play with large language models has emerged as a promising paradigm for achieving self-improving artificial intelligence. However, existing self-play frameworks often suffer from optimization instability, due to (i) non-stationary…
We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We…
We introduce a system of software tools that can automatically prove or discover geometric inequalities. The system, called GeoGebra Discovery, consisting of an extended version of GeoGebra, a controller web service realgeom, and the…
Compared with flatbed scanners, portable smartphones provide more convenience for physical document digitization. However, such digitized documents are often distorted due to uncontrolled physical deformations, camera positions, and…
In this paper, we propose a new Deep Neural Network (DNN) testing algorithm called the Constrained Gradient Descent (CGD) method, and an implementation we call CGDTest aimed at exposing security and robustness issues such as adversarial…