Semantic Representations of Mathematical Expressions in a Continuous Vector Space

Mathematical notation makes up a large portion of STEM literature, yet, finding semantic representations for formulae remains a challenging problem. Because mathematical notation is precise, and its meaning changes significantly with small character shifts, the methods that work for natural text do not necessarily work well for mathematical expressions. In this work, we describe an approach for representing mathematical expressions in a continuous vector space. We use the encoder of a sequence-to-sequence architecture, trained on visually different but mathematically equivalent expressions, to generate vector representations (or embeddings). We compare this approach with an autoencoder and show that the former is better at capturing mathematical semantics. Finally, to expedite future research, we publish a corpus of equivalent transcendental and algebraic expression pairs.Comment: 17 pages, 2 figure

Modeling of magnetization dynamics and applications to spin-based logic and memory devices

The objective of this research is to develop models to better evaluate the performance and reliability of proposed spin-based boolean devices. This research will focus on a particular spin-based logic technology called Spin-Switch Logic. There are two primary reversal mechanisms that will be considered for a full evaluation of Spin-Switch technology. Firstly, nanomagnet reversal through the use of spin-transfer torque (STT) is studied. While switching through STT has been analytically solved for the uniaxial nanomagnet case, the biaxial case has yet to be studied on a sufficient scale and will be a focus of this research. Secondly, input-output isolation is achieved through dipolar coupling; hence, the performance and reliability of this type of reversal mechanism is extensively studied. It is shown that dipolar coupling strength is not only a function of geometric and material parameters, but also of reversal speed. If the reversal of a neighboring nanomagnet is very fast, the dipolar field reduces to a constant longitudinal field and can be analytically studied. However, if the reversal of the neighboring nanomagnet is slow, new models are needed to estimate the region of reliable coupling and delay. Lastly, a focal point of this research will be on the reliability of nanomagnet states in the presence of thermal noise and new models are proposed to estimate the reliability of complex spin-based systems. Not only does the thermal noise affect the probability of magnetization state consistency, it also alters nanomagnet precession during reversal, making the delay a random variable. Hence, models are developed for evaluating the variation in reversal delay through STT for both uniaxial and biaxial cases. Ultimately, these analytic models are combined to comprehensively evaluate the performance of Spin-Switch technology and identify possible improvements to this technology. While the end result of this research will be a thorough analysis of Spin-Switch logic, the models developed during this research are applicable to a variety of spin-based logic and memory technologies.Ph.D

Highlighting Named Entities in Input for Auto-Formulation of Optimization Problems

Operations research deals with modeling and solving real-world problems as mathematical optimization problems. While solving mathematical systems is accomplished by analytical software, formulating a problem as a set of mathematical operations has been typically done manually by domain experts. However, recent machine learning models have shown promise in converting textual problem descriptions to corresponding mathematical formulations. In this paper, we present an approach that converts linear programming word problems into meaning representations that are structured and can be used by optimization solvers. Our approach uses the named entity-based enrichment to augment the input and achieves state-of-the-art accuracy, winning the second task of the NL4Opt competition (https://nl4opt.github.io).Comment: 5 pages, 4 figures, 1 tabl