Exel's crossed product for non-unital C*-algebras

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for \alpha. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.Comment: 22 page

A Simple Theory of Every 'Thing'

One of the criteria to a strong principle in natural sciences is simplicity. This paper claims that the Free Energy Principle (FEP), by virtue of unifying particles with mind, is the simplest. Motivated by Hilbert’s 24th problem of simplicity, the argument is made that the FEP takes a seemingly mathematical complex domain and reduces it to something simple. More specifically, it is attempted to show that every ‘thing’, from particles to mind, can be partitioned into systemic states by virtue of self-organising symmetry break, i.e. self-entropy in terms of the balance between risk and ambiguity to achieve epistemic gain. By virtue of its explanatory reach, the FEP becomes the simplest principle under quantum, statistical and classical mechanics conditions

Machine Learning and the Future of Realism

The preceding three decades have seen the emergence, rise, and proliferation of machine learning (ML). From half-recognised beginnings in perceptrons, neural nets, and decision trees, algorithms that extract correlations (that is, patterns) from a set of data points have broken free from their origin in computational cognition to embrace all forms of problem solving, from voice recognition to medical diagnosis to automated scientific research and driverless cars, and it is now widely opined that the real industrial revolution lies less in mobile phone and similar than in the maturation and universal application of ML. Among the consequences just might be the triumph of anti-realism over realism

A Dynamical Systems Approach for Static Evaluation in Go

In the paper arguments are given why the concept of static evaluation has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling static evaluation through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated.Comment: IEEE Transactions on Computational Intelligence and AI in Games, vol 3 (2011), no

Ideal structure of crossed products by endomorphisms via reversible extensions of C∗C^*-dynamical systems

We consider an extendible endomorphism $\alpha$ of a $C^*$-algebra $A$. We associate to it a canonical $C^*$-dynamical system $(B,\beta)$ that extends $(A,\alpha)$ and is `reversible' in the sense that the endomorphism $\beta$ admits a unique regular transfer operator $\beta_*$. The theory for $(B,\beta)$ is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for $(B,\beta)$ in terms of the initial system $(A,\alpha)$. We apply this idea to study the ideal structure of a non-unital version of the crossed product $C^*(A,\alpha,J)$ introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal $J$ in $(\ker\alpha)^\bot$, and if $J=(\ker\alpha)^\bot$ it is a modification of Stacey's crossed product that works well with non-injective $\alpha$'s. We provide descriptions of the lattices of ideals in $C^*(A,\alpha,J)$ consisting of gauge-invariant ideals and ideals generated by their intersection with $A$. We investigate conditions under which these lattices coincide with the set of all ideals in $C^*(A,\alpha,J)$. In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of $\alpha$ or pointwise quasinilpotence of $\alpha$.Comment: 35 pages, Appendix on $C^*(A,\alpha,J)$ viewed as relative Cuntz-Pimsner algebras is added, this version is accepted to Internat. J. Mat

Parameter estimation for Boolean models of biological networks

Boolean networks have long been used as models of molecular networks and play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.Comment: Web interface of the software is available at http://polymath.vbi.vt.edu/polynome