Related papers: Umbral Calculus, a Different Mathematical Language
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie…
We define an extension of lambda-calculus with dependents types that enables us to encode transparent and opaque probabilistic programs and prove a strong normalisation result for it by a reducibility technique. While transparent…
Declarative modeling uses symbolic expressions to represent models. With such expressions one can formalize high-level mathematical computations on models that would be difficult or impossible to perform directly on a lower-level simulation…
While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a…
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
Passive documents and active programs now widely comingle. Document languages include Turing-complete programming elements, and programming languages include sophisticated document notations. However, there are no formal foundations that…
In this paper, we present a new approach to the semantic enrichment of mathematical expression problem. Our approach is a combination of statistical machine translation and disambiguation which makes use of surrounding text of the…
In this paper, we utilize operational methods to obtain closed-form solutions for certain classes of integrals in the spirit of Ramanujan's Master Theorem and provide several analogs to it. Although the use of operational calculus makes the…
A unified mathematical language for medicine and science will be presented. Using this language, models for DNA replication, protein synthesis, chemical reactions, neurons and a cardiac cycle of a heart have been built. Models for Turing…
In the former article "Formal mathematical systems including a structural induction principle" we have presented a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the…
We present a logical separability analysis for a functional quantum computation language. This logic is inspired by previous works on logical analysis of aliasing for imperative functional programs. Both analyses share similarities notably…
Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic…
The Unified Modelling Language is emerging as a de-facto standard for modelling object-oriented systems. However, the semantics document that a part of the standard definition primarily provides a description of the language's syntax and…
Autoformalization has emerged as a term referring to the automation of formalization - specifically, the formalization of mathematics using interactive theorem provers (proof assistants). Its rapid development has been driven by progress in…
We study planning with submodular objective functions, where instead of maximizing the cumulative reward, the goal is to maximize the objective value induced by a submodular function. Our framework subsumes standard planning and submodular…
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.
In the quest to give a formal compositional semantics to natural languages, semanticists have started turning their attention to phenomena that have been also considered as parts of pragmatics (e.g., discourse anaphora and presupposition…
In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.
The lower and upper bound of any given algorithm is one of the most crucial pieces of information needed when evaluating the computational effectiveness for said algorithm. Here a novel method of Boolean Algebraic Programming for symbolic…
A plausible definition of "reasoning" could be "algebraically manipulating previously acquired knowledge in order to answer a new question". This definition covers first-order logical inference or probabilistic inference. It also includes…