Martin Gardner and His Influence on Recreational Math

Recreational mathematics is a relatively new field in the world of mathematics. While sometimes overlooked as frivolous since those who practice it need no advanced knowledge of the subject, recreational mathematics is a perfect transition for people to experience the joy in logically establishing a solution. Martin Gardner recognized that this pattern of proving solutions to questions is how mathematics progresses. From his childhood on, Gardner greatly influenced the mathematical world. Although not a mathematician, he inspired many to pursue careers and make advancements in mathematics during his 25-year career with Scientific American. He encouraged novices to expand their knowledge, enlightened professionals of computer science developments, and established his own proofs

Intelligent systems in manufacturing: current developments and future prospects

Global competition and rapidly changing customer requirements are demanding increasing changes in manufacturing environments. Enterprises are required to constantly redesign their products and continuously reconfigure their manufacturing systems. Traditional approaches to manufacturing systems do not fully satisfy this new situation. Many authors have proposed that artificial intelligence will bring the flexibility and efficiency needed by manufacturing systems. This paper is a review of artificial intelligence techniques used in manufacturing systems. The paper first defines the components of a simplified intelligent manufacturing systems (IMS), the different Artificial Intelligence (AI) techniques to be considered and then shows how these AI techniques are used for the components of IMS

The limits to prediction in ecological systems

Predicting the future trajectories of ecological systems is increasingly important as the magnitude of anthropogenic perturbation of the earth systems grows.We distinguish between two types of predictability: the intrinsic or theoretical predictability of a system and the realized predictability that is achieved using available models and parameterizations. We contend that there are strong limits on the intrinsic predictability of ecological systems that arise from inherent characteristics of biological systems. While the realized predictability of ecological systems can be limited by process and parameter misspecification or uncertainty, we argue that the intrinsic predictability of ecological systems is widely and strongly limited by computational irreducibility. When realized predictability is low relative to intrinsic predictability, prediction can be improved through improved model structure or specification of parameters. Computational irreducibility, however, asserts that future states of the system cannot be derived except through computation of all of the intervening states, imposing a strong limit on the intrinsic or theoretical predictability. We argue that ecological systems are likely to be computationally irreducible because of the difficulty of pre-stating the relevant features of ecological niches, the complexity of ecological systems and because the biosphere can enable its own novel system states or adjacent possible. We argue that computational irreducibility is likely to be pervasive and to impose strong limits on the potential for prediction in ecology. Copyright

Shortest Loops are Pacemakers in Random Networks of Electrically Coupled Axons

High-frequency oscillations (HFOs) are an important part of brain activity in health and disease. However, their origins remain obscure and controversial. One possible mechanism depends on the presence of sparsely distributed gap junctions that electrically couple the axons of principal cells. A plexus of electrically coupled axons is modeled as a random network with bi-directional connections between its nodes. Under certain conditions the network can demonstrate one of two types of oscillatory activity. Type I oscillations (100–200 Hz) are predicted to be caused by spontaneously spiking axons in a network with strong (high conductance) gap junctions. Type II oscillations (200–300 Hz) require no spontaneous spiking and relatively weak (low-conductance) gap junctions, across which spike propagation failures occur. The type II oscillations are reentrant and self-sustained. Here we examine what determines the frequency of type II oscillations. Using simulations we show that the distribution of loop lengths is the key factor for determining frequency in type II network oscillations. We first analyze spike failure between two electrically coupled cells using a model of anatomically reconstructed CA1 pyramidal neuron. Then network oscillations are studied by a cellular automaton model with random network connectivity, in which we control loop statistics. We show that oscillation periods can be predicted from the network’s loop statistics. The shortest loop, around which a spike can travel, is the most likely pacemaker candidate. The principle of one loop as a pacemaker is remarkable, because random networks contain a large number of loops juxtaposed and superimposed, and their number rapidly grows with network size. This principle allows us to predict the frequency of oscillations from network connectivity and visa versa. We finally propose that type I oscillations may correspond to ripples, while type II oscillations correspond to so-called fast ripples

Information Flow in the Spatiotemporal Dynamics of Cellular Automata

Decision making in natural systems, such as the body\u27s immune response to a potential pathogen or a bacterial colony\u27s initiation of fruiting due to food scarcity, is distributed over many cells that posses only local information, and not determined globally. Understanding how accurate decisions can be made in such systems where no individual decisions maker has complete information has important implications in distributed software and can provide insights into the biological evolution of complexity. In this work, the process of distributed decision making is modeled using the majority problem in cellular automata, and information theoretic measures of Kolmogorov complexity are applied to quantify information flow during the decision making process. Results show that (a) when the decision making process converges the information content of the dynamics quickly reaches a peak then decays to near-zero; (b) if the process does not converge and becomes chaotic, information content oscillates over a large unstable range; (c) extensive statistically significant differences exist in information flow dynamics between convergent and chaotic outcomes; and (d) there are small, but statistically significant differences in information flow dynamics between convergence to the incorrect answer. This last result supports the hypothesis that correct decision making maximized information flow among agents in distributed decision making

Cellular Automata

Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented

Probabilistic Deterministic Finite Automata and Recurrent Networks, Revisited

Reservoir computers (RCs) and recurrent neural networks (RNNs) can mimic any finite-state automaton in theory, and some workers demonstrated that this can hold in practice. We test the capability of generalized linear models, RCs, and Long Short-Term Memory (LSTM) RNN architectures to predict the stochastic processes generated by a large suite of probabilistic deterministic finite-state automata (PDFA). PDFAs provide an excellent performance benchmark in that they can be systematically enumerated, the randomness and correlation structure of their generated processes are exactly known, and their optimal memory-limited predictors are easily computed. Unsurprisingly, LSTMs outperform RCs, which outperform generalized linear models. Surprisingly, each of these methods can fall short of the maximal predictive accuracy by as much as 50% after training and, when optimized, tend to fall short of the maximal predictive accuracy by ~5%, even though previously available methods achieve maximal predictive accuracy with orders-of-magnitude less data. Thus, despite the representational universality of RCs and RNNs, using them can engender a surprising predictive gap for simple stimuli. One concludes that there is an important and underappreciated role for methods that infer "causal states" or "predictive state representations"

Enabling collaborative numerical modeling in earth sciences using knowledge infrastructure

Knowledge Infrastructure is an intellectual framework for creating, sharing, and distributing knowledge. In this paper, we use Knowledge Infrastructure to address common barriers to entry to numerical modeling in Earth sciences: computational modeling education, replicating published model results, and reusing published models to extend research. We outline six critical functional requirements: 1) workflows designed for new users; 2) a community-supported collaborative web platform; 3) distributed data storage; 4) a software environment; 5) a personalized cloud-based high-performance computing platform; and 6) a standardized open source modeling framework. Our methods meet these functional requirements by providing three interactive computational narratives for hands-on, problem-based research demonstrating how to use Landlab on HydroShare. Landlab is an open-source toolkit for building, coupling, and exploring two-dimensional numerical models. HydroShare is an online collaborative environment for the sharing of data and models. We describe the methods we are using to accelerate knowledge development by providing a suite of modular and interoperable process components that allows students, domain experts, collaborators, researchers, and sponsors to learn by exploring shared data and modeling resources. The system is designed to support uses on the continuum from fully-developed modeling applications to prototyping research software tools