Positive Alexander Duality for Pursuit and Evasion

Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive (co)homology generalized to arbitrary degrees; Positive Alexander Duality generalized from homological degrees 0,1; Morse and smoothness conditions generalized; illustrations of positive homology added. minor corrections in proofs, notation, organization, and language made throughout. variant of Borel-Moore homology now use

Cyclic Cellular Automata on Networks and Cohomological Waves

A dynamic coverage problem for sensor networks that are sufficiently dense but not localized is considered. By maintaining only a small fraction of sensors on at any time, we are aimed to find a decentralized protocol for establishing dynamic, sweeping barriers of awake-state sensors. Network cyclic cellular automata is used to generate waves. By rigorously analyzing network-based cyclic cellular automata in the context of a system of narrow hallways, it shows that waves of awake-state nodes turn corners and automatically solve pusuit/evasion-type problems without centralized coordination. As a corollary of this work, we unearth some interesting topological interpretations of features previously observed in cyclic cellular automata (CCA). By considering CCA over networks and completing to simplicial complexes, we induce dynamics on the higher-dimensional complex. In this setting, waves are seen to be generated by topological defects with a nontrivial degree (or winding number). The simplicial complex has the topological type of the underlying map of the workspace (a subset of the plane), and the resulting waves can be classified cohomologically. This allows one to program pulses in the sensor network according to cohomology class. We give a realization theorem for such pulse waves

Stepping Beyond the Newtonian Paradigm in Biology. Towards an Integrable Model of Life: Accelerating Discovery in the Biological Foundations of Science

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses

This Week's Finds in Mathematical Physics (1-50)

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).Comment: 242 page

Report / Institute für Physik

In this report the Institutes of Physics of the Universität Leipzig present their scientific activities and major achievements in the year 2003

Reports to the President

A compilation of annual reports including a report from the President of the Massachusetts Institute of Technology, as well as reports from the academic and administrative units of the Institute. The reports outline the year's goals, accomplishments, honors and awards, and future plans

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum