From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity

Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do with our understanding of randomness, irreversibility and unpredictability using physical theory, and these in turn undermine our certainty regarding what we can and what we cannot know about complex phenomena in general. The sources of complexity examined herein appear to be channels for the amplification of naturally occurring randomness in the physical world. Our analysis suggests that when the conditions for the renormalization group apply, this spontaneous randomness, which is not a reflection of our limited knowledge, but a genuine property of nature, does not realize the conventional thermodynamic state, and a new condition, intermediate between the dynamic and the thermodynamic state, emerges. We argue that with this vision of complexity, life, which with ordinary statistical mechanics seems to be foreign to physics, becomes a natural consequence of dynamical processes.Comment: Phylosophica

Remarks on the analyticity of subadditive pressure for products of triangular matrices

We study Falconer's subadditive pressure function with emphasis on analyticity. We begin by deriving a simple closed form expression for the pressure in the case of diagonal matrices and, by identifying phase transitions with zeros of Dirichlet polynomials, use this to deduce that the pressure is piecewise real analytic. We then specialise to the iterated function system setting and use a result of Falconer and Miao to extend our results to include the pressure for systems generated by matrices which are simultaneously triangularisable. Our closed form expression for the pressure simplifies a similar expression given by Falconer and Miao by reducing the number of equations needing to be solved by an exponential factor. Finally we present some examples where the pressure has a phase transition at a non-integer value and pose some open questions.Comment: 10 pages, 1 figure, to appear in Monatshefte f\"ur Mathemati

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Cuntz–Krieger Algebras and Wavelets on Fractals

We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets

Cuntz-Krieger algebras and wavelets on fractals

We consider representations of Cuntz--Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron-Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.Comment: 37 pages, LaTe