Related papers: A cute proof that makes $e$ natural
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
Geometrically, $\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve $\frac{1}{x}$ from $a$ to $b$, where $0<a<b$, and this area gives a positive number. Using this area argument, in this expository note, we present some visual…
The study examines the relationship between Ball's magic numbers and reverses divisors. These numbers are the source of beautiful and curious properties. Activities related to numbers can be a fun way to motivate mathematics students, while…
A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
We analyse preference inference, through consistency, for general preference languages based on lexicographic models. We identify a property, which we call strong compositionality, that applies for many natural kinds of preference…
This note describes a way of obtaining e that differs from the standard one. It could be used as an alternate way of showing how the value of e is obtained. No attempt is made to show the existence of the limit in the definition of e that…
This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $\lambda$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using…
This is a kind of survey on properties of correlations of two very general arithmetic functions, mainly from the point of view of Ramanujan expansions. In fact, our previous papers on these links had, as a focus, the "Ramanujan…
E prover is a state-of-the-art theorem prover for first-order logic with equality. E prover is built around a saturation loop, where new clauses are derived by inference rules from previously derived clauses. Selection of clauses for the…
The hypergraph product (HGP) is a famous code construction technique with an equally famous canonical visualisation. This visual perspective provides much more than simply a way to build intuition: HGP codes can be defined graphically,…
In arXiv:1405.4527 Connes and Consani introduced and studied the arithmetic site and showed that the isomorphism classes of points are in canonical bijection with the finite adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}}…
Logical relations (LR) have been around for many years, and today they are used in many formal results. However, it can be difficult to LR beginners to find a good place to start to learn. Papers often use highly specialized LRs that use…
In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.
This report first shows the equivalence bewteen several formulations of classical logic in intuitionistic logic (tertium non datur, reductio ad absurdum, Pierce's law). Then it establishes the correctness of the G\"odel-Kolmogorov…
A theoretical self-sustainable economic model is established based on the fundamental factors of production, consumption, reservation and reinvestment, where currency is set as a unconditional credit symbol serving as transaction equivalent…
In order to describe natural phenomena, science develops sophisticated models that use mathematical and formal languages which seem, and often are, very far from common experience. When a phenomenon is not accessible to our senses, its…
This work showcases a new approach for causal discovery by leveraging user experiments and recent advances in photo-realistic image editing, demonstrating a potential of identifying causal factors and understanding complex systems…
We give a new, elementary proof of what we believe is the simplest known example of a ``natural'' problem in computational 3-dimensional topology that is $\mathsf{NP}$-hard -- namely, the \emph{Trivial Sublink Problem}: given a diagram $L$…
Environmental Education (EE) is vital for shaping citizens who understand and value sustainability as an epistemological and practical alternative to mitigate current environmental issues. This research was prompted by the exploration of…