Mathematical Foundations of Consciousness

We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to frame mathematical foundations of consciousness. To do this we introduce a supplementary axiomatic system that characterizes experience and consciousness as primitives. The new axioms proceed through characterization of so- called consciousness operators. The Russell operator plays a central role and is shown to be one example of a consciousness operator. Neural networks supply striking examples of non-well-founded graphs the decorations of which generate associated sets, each with a Platonic aspect. Employing our foundations, we show how the supervening of consciousness on its neural correlates in the brain enables the framing of a theory of consciousness by applying appropriate consciousness operators to the generated sets in question

Modeling the clonal heterogeneity of stem cells

Recent experimental studies suggest that tissue stem cell pools are composed of functionally diverse clones. Metapopulation models in ecology concentrate on collections of populations and their role in stabilizing coexistence and maintaining selected genetic or epigenetic variation. Such models are characterized by expansion and extinction of spatially distributed populations. We develop a mathematical framework derived from the multispecies metapopulation model of Tilman et al (1994) to study the dynamics of heterogeneous stem cell metapopulations. In addition to normal stem cells, the model can be applied to cancer cell populations and their response to treatment. In our model disturbances may lead to expansion or contraction of cells with distinct properties, reflecting proliferation, apoptosis, and clonal competition. We first present closed-form expressions for the basic model which defines clonal dynamics in the presence of exogenous global disturbances. We then extend the model to include disturbances which are periodic and which may affect clones differently. Within the model framework, we propose a method to devise an optimal strategy of treatments to regulate expansion, contraction, or mutual maintenance of cells with specific properties

FAST HYBRID SOLUTION OF ALGEBRAIC SYSTEMS

We propose and analyze the error and timing of solvers consisting of both analog and digital circuitry for sparse linear systems of equations. We obtain high speed, but low precision from the analog circuits. We combine this with low speed, but high precision from the digital circuits. The hybrid circuit should be faster than digital circuits alone. As a preconditioner to standard iterative solution methods, the hybrid circuit makes the cost of the preconditioning step negligible. We also apply the hybrid circuit to a standard multilevel algorithm

BEYOND MASSIVE PARALLELISM: NUMERICAL COMPUTATION USING ASSOCIATIVE TABLES

Novel computing devices are exploited for numerical computation. The solution of a numerical problem is sought, which has been solved many times before, but this time with a different set of input data. A table is a classical way to collect the old solutions in order to exploit them to nd the new one. This process is extended to more general problems than the usual function value approximation. To do this, a new concept of table is introduced. These tables are addressed associatively. Several problems are treated both theoretically and computationally. These problems include solving linear systems of equations, partial differential equations, nonlinear systems of ordinary differential equations, and Karmarkar's algorithm. Hardware requirements are discussed