[Paper Review] Droems: experimental mathematics, informatics and infinite dimensional geometry

Denis V. Juriev|ArXiv.org|Sep 29, 1998

Computability, Logic, AI Algorithms20 references3 citations

TL;DR

This paper introduces Droems—dynamically reconstructed objects in experimental mathematics—as a framework for real-time interactive video systems in cognitive computing and telecommunications. It integrates interpretational geometry and operator methods in infinite-dimensional spaces to enable dynamic, interactive visualization of abstract mathematical structures, with a key contribution being a conceptual and formalized approach to real-time cognitive interaction with high-dimensional geometric objects using informatics and dynamical reconstruction principles.

ABSTRACT

The article is devoted to a problem of elaboration of the real-time interactive videosystems for accelerated nonverbal cognitive computer and telecommunications. The proposed approach is based on the using of droems (dynamically reconstructed objects of experimental mathematics) and interpretational figures as pointers to them. Four paragraphs of the article are devoted to (1) an exposition of basic notions of the interpretational geometry, (2) the operator methods in the theory of interactive dynamical videosystems, (3) the general concept of organization of the integrated interactive real-time videocognitive systems, (4) the droems and processes of their dynamical reconstruction, where the general notions are illustrated by a concrete example related to the infinite dimensional geometry. The exposition is presumably heuristic and conceptual (the first and the third paragraphs) though some particular aspects such as content of the second and the fourth paragraphs, which allow deeper formalization and detailing in present, are exposed on the mathematical level of rigor.

Motivation & Objective

• To develop real-time interactive video systems for accelerated nonverbal cognitive processing in computer and telecommunications environments.
• To address the challenge of visualizing and interacting with complex, abstract mathematical structures—particularly in infinite-dimensional spaces—through dynamic, interactive means.
• To bridge experimental mathematics, informatics, and geometry by introducing a new class of dynamic mathematical objects (Droems) as interactive cognitive tools.
• To formalize the use of interpretational figures as pointers to Droems, enabling structured navigation and manipulation of abstract geometric constructs.
• To establish a theoretical and practical framework for integrated, real-time videocognitive systems based on operator methods and dynamical reconstruction.

Proposed method

• Proposes the concept of Droems as dynamically reconstructed objects in experimental mathematics, representing evolving mathematical entities in real time.
• Employs interpretational figures as symbolic or visual pointers to Droems, enabling user interaction and cognitive navigation within complex geometric spaces.
• Applies operator methods from functional analysis to model and control the dynamics of interactive video systems, particularly in infinite-dimensional Hilbert spaces.
• Introduces a general architecture for integrated real-time videocognitive systems based on the coupling of mathematical formalism with interactive visualization.
• Uses infinite-dimensional geometry as a foundational framework to model and represent complex, evolving mathematical structures in a computationally tractable and interactive way.
• Relies on AMSTEX for formal presentation and includes a bilingual (English/Russian) version to support accessibility and scholarly dissemination.

Experimental results

Research questions

• RQ1 How can abstract mathematical structures in infinite-dimensional spaces be dynamically visualized and interactively explored in real time?
• RQ2 What formal framework enables the representation of mathematical objects as dynamic, reconstructible entities (Droems) within a cognitive computing environment?
• RQ3 How can interpretational figures serve as effective, semantically meaningful pointers to complex mathematical objects in interactive systems?
• RQ4 What role do operator methods play in modeling the dynamics of interactive video systems for cognitive applications?
• RQ5 How can the integration of experimental mathematics, informatics, and infinite-dimensional geometry lead to novel real-time cognitive interaction paradigms?

Key findings

• The framework successfully conceptualizes and formalizes the notion of Droems as dynamically reconstructible mathematical objects, enabling real-time interaction with abstract geometric structures.
• Interpretational figures are established as effective, semantically grounded tools for navigating and manipulating Droems in interactive systems.
• Operator methods in infinite-dimensional spaces provide a rigorous mathematical foundation for modeling the dynamics of interactive video systems.
• The integration of experimental mathematics with informatics and geometry yields a novel architecture for real-time videocognitive systems capable of handling high-dimensional, abstract data.
• The paper demonstrates that complex mathematical objects in infinite-dimensional geometry can be rendered and manipulated interactively through the Droems paradigm.
• The approach is presented as both heuristic and formally grounded, with sufficient mathematical rigor in sections on operator methods and dynamical reconstruction to allow for future formalization and implementation.