math.HO

arXiv:2503.10320v1 Announce Type: cross Abstract: Cellular Automata (CA) are commonly investigated as a particular type of dynamical systems, defined by shift-invariant local rules. In this paper, we consider instead CA as algebraic systems, focusing on the combinatorial designs induced by their short-term behavior. Specifically, we review the main results published in the literature concerning the construction of mutually orthogonal Latin squares via bipermutive CA, considering both the linear and nonlinear cases. We then survey some significant applications of these results to cryptography, and conclude with a discussion of open problems to be addressed in future research on CA-based combinatorial designs.

arXiv:2409.17304v2 Announce Type: replace-cross Abstract: Signal Processing (SP) and Machine Learning (ML) rely on good math and coding knowledge, in particular, linear algebra, probability, trigonometry, and complex numbers. A good grasp of these relies on scalar algebra learned in middle school. The ability to understand and use scalar algebra well, in turn, relies on a good foundation in basic arithmetic. Because of various systemic barriers, many students are not able to build a strong foundation in arithmetic in elementary school. This leads them to struggle with algebra and everything after that. Since math learning is cumulative, the gap between those without a strong early foundation and everyone else keeps increasing over the school years and becomes difficult to fill in college. In this article we discuss how SP faculty, students, and professionals can play an important role in starting, and participating in, university-run, or other, out-of-school math support programs to supplement students' learning. Two example programs run by the authors, CyMath at Iowa State and Algebra by 7th Grade (Ab7G) at Purdue, and one run by the Actuarial Foundation, are described. We conclude with providing some simple zero-cost suggestions for public schools that, if adopted, could benefit a much larger number of students than what out-of-school programs can reach.

arXiv:2307.09998v5 Announce Type: replace Abstract: This paper investigates how hallucination rates in Large Language Models (LLMs) may be controlled via a symbolic data generation framework, exploring a fundamental relationship between the rate of certain mathematical errors and types of input intervention. Specifically, we systematically generate data for a derivation generation task using a symbolic engine, applying targeted interventions to prompts to perturb features of mathematical derivations such as the surface forms of symbols, equational tree structures, and mathematical context. We then evaluate the effect of prompt interventions across a range of LLMs including fine-tuned T5 models, GPT, and LLaMa-based models. Our experiments suggest that T5-Large can outperform the few-shot performance of GPT-4 on various evaluation sets generated via the framework. However, an extensive evaluation based on human analysis, template-based error detection, and text generation metrics reveals model weaknesses beyond what the reference-based metrics singularly describe. We use these results to tie characteristic distributional footprints of interventions to the human evaluation of LLM derivation quality, potentially leading to significant control over fine-grained mathematical capabilities of language models with respect to specific types of errors.

arXiv:2501.01433v1 Announce Type: new Abstract: While logic puzzles have engaged individuals through problem-solving and critical thinking, the creation of new puzzle rules has largely relied on ad-hoc processes. Pencil puzzles, such as Slitherlink and Sudoku, represent a prominent subset of these games, celebrated for their intellectual challenges rooted in combinatorial logic and spatial reasoning. Despite extensive research into solving techniques and automated problem generation, a unified framework for systematic and scalable rule design has been lacking. Here, we introduce a mathematical framework for defining and systematizing pencil puzzle rules. This framework formalizes grid elements, their positional relationships, and iterative composition operations, allowing for the incremental construction of structures that form the basis of puzzle rules. Furthermore, we establish a formal method to describe constraints and domains for each structure, ensuring solvability and coherence. Applying this framework, we successfully formalized the rules of well-known Nikoli puzzles, including Slitherlink and Sudoku, demonstrating the formal representation of a significant portion (approximately one-fourth) of existing puzzles. These results validate the potential of the framework to systematize and innovate puzzle rule design, establishing a pathway to automated rule generation. By providing a mathematical foundation for puzzle rule creation, this framework opens avenues for computers, potentially enhanced by AI, to design novel puzzle rules tailored to player preferences, expanding the scope of puzzle diversity. Beyond its direct application to pencil puzzles, this work illustrates how mathematical frameworks can bridge recreational mathematics and algorithmic design, offering tools for broader exploration in logic-based systems, with potential applications in educational game design, personalized learning, and computational creativity.

arXiv:2501.00002v1 Announce Type: new Abstract: In this paper we present a QUBO formulation for the Takuzu game (or Binairo), for the most recent LinkedIn game, Tango, and for its generalizations. We optimize the number of variables needed to solve the combinatorial problem, making it suitable to be solved by quantum devices with fewer resources.

arXiv:2412.17265v1 Announce Type: new Abstract: Xiaomai is an intelligent tutoring system (ITS) designed to help Chinese college students in learning advanced mathematics and preparing for the graduate school math entrance exam. This study investigates two distinctive features within Xiaomai: the incorporation of free-response questions with automatic feedback and the metacognitive element of reflecting on self-made errors.

arXiv:2412.17140v1 Announce Type: cross Abstract: The presented work focuses on problems from determinant theory, set theory and topology. The term graph is the binding element that connects these problems. Graphs are distinguished by their geometrical simplicity, which helps in showing the equivalence between various seemingly unrelated problems, besides providing solutions to several open questions discussed here.

arXiv:2412.15436v1 Announce Type: cross Abstract: This memorial article for Mark Sapir provides a brief overview of his life and career. Among his many contributions we highlight two of his most celebrated achievements: his groundbreaking solutions to Burnside-type problems for semigroups and his innovative construction of S-machines. Additionally, reflections from his colleagues and friends offer a heartfelt tribute, blending professional insights with personal memories.