Related papers: From Euler to AI: Unifying Formulas for Mathematic…
Quantification is the machine learning task of estimating test-data class proportions that are not necessarily similar to those in training. Apart from its intrinsic value as an aggregate statistic, quantification output can also be used to…
In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…
Handwritten Mathematical Expression Recognition (HMER) remains a persistent challenge in Optical Character Recognition (OCR) due to the inherent freedom of symbol layouts and variability in handwriting styles. Prior methods have faced…
The challenge of information extraction (IE) lies in the diversity of label schemas and the heterogeneity of structures. Traditional methods require task-specific model design and rely heavily on expensive supervision, making them difficult…
Benchmarking the capabilities of AI systems, including Large Language Models (LLMs) and Vision Models, typically ignores the impact of uncertainty in the underlying ground truth answers from experts. This ambiguity is not just limited to…
Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of…
The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a…
Many robotics applications require alignment and fusion of observations obtained at multiple views to form a global model of the environment. Multi-way data association methods provide a mechanism to improve alignment accuracy of pairwise…
Reasoning on large and complex real-world models is a computationally difficult task, yet one that is required for effective use of many AI applications. A plethora of inference algorithms have been developed that work well on specific…
This paper studies the unification problem with associative, commutative, and associative-commutative functions mainly from a viewpoint of the parameterized complexity on the number of variables. It is shown that both associative and…
A hallmark of intelligence is the ability to use a familiar domain to make inferences about a less familiar domain, known as analogical reasoning. In this article, we delve into the performance of Large Language Models (LLMs) in dealing…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
Building robust, interpretable, and secure AI system requires quantifying and representing uncertainty under a probabilistic perspective to mimic human cognitive abilities. However, probabilistic computation presents significant challenges…
Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this…
Uniform interpolation property (UIP) is a strengthening of Craig interpolation property. It can be understood as the definability of propositional quantifiers. This paper develops the sequent calculi provided in Murai and Sano (2020),…
Underlying data distributions of natural language, programming code, and mathematical symbols vary vastly, presenting a complex challenge for large language models (LLMs) that strive to achieve high performance across all three domains…
The philosophy of mathematical practice (PMP) looks to evidence from working mathematics to help settle philosophical questions. One prominent program under the PMP banner is the study of explanation in mathematics, which aims to understand…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…
With the development of artificial intelligence (AI), large language models (LLM) are widely used in many fields. However, the reasoning ability of LLM is still very limited when it comes to mathematical reasoning. Mathematics plays an…
By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we…