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Understanding realistic complex systems requires confronting significant conceptual, theoretical and experimental limitations rooted in the persistence of views that originated in the mechanics of simple moving bodies. We define the category of complex multiscale stochastic systems as a useful device for capturing the minimally required complexity of many types of phenomena. In doing so, we provide evidence indicating that determinism, continuity and reversibility can lead to theoretical inadequacies that manifest as intractability, inaccuracies or non-representativeness when applied to complex systems. We take the view that interactions are fundamental and summarize their portrayal across many disciplines. Despite their centrality, interactions remain largely neglected as subjects of research interest of their own. We hypothesize that a generalized theory of interactions may help organize evidence from multiple scientific domains towards a more unified realistic view of systems.

In this work the generalized Collatz problem $qn+1$ ($q$ odd) is studied. As a natural generalization of the original $3n+1$ problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number theory and dynamical systems, general properties are established, such as the existence of finitely many periodic sequences for each $q$. In particular, when $q$ is a Mersenne number, $q=2^p-1$, there only exists one such cycle, known as the trivial one. Further analysis based on a probabilistic model shows that for $q=3$ the asymptotic behavior of all sequences is always convergent, whereas for $q\geq 5$ the asymptotic behavior of the sequences is divergent for almost all numbers (for a set of natural density one). This leads to the conclusion that the so called Collatz Conjecture is true, and that $q=3$ is a very special case among the others (Crandall conjecture). Indeed, it is conjectured that the general problem $qn+1$ is undecidable., Comment: 26 pages. Preprint. Submitted for a peer-review process. Comments on v3: Theorem 4 has been updated from q-1 possible cycles to finitely many possible cycles. The proof follows a new argument. Theorem 5 has a shorter proof and a separated lemma, and the proof of proposition 14 (now proposition 16) has been fixed. It is also included a new reference

This study investigate the effectiveness of adding an "explicit general problem solving teaching strategy" (EGPS) to guided inquiry (GI) on pre-service elementary school teachers' ability to solve heat transfer problems. The pre-service elementary teachers in this study were enrolled in two sections of a chemistry course for pre-service elementary teachers' at a four year university. One section of this class was taught a treatment method, the explicit general problem solving approach, while the other section was taught using guided inquiry approach. Quantitative data was obtained using a post-test while qualitative data was obtained probing questions. The scores for the two teaching approaches showed no significant difference. Further, qualitative data indicated little difference in the way participants solved their problems regardless of instructional approach. Hence, this study concludes that both EGPS and guided inquiry have comparable effectiveness of teaching problem solving.

This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are interpreted as states and morphisms as transformations between states. Moreover, cognitive problems are reduced to the specification of two objects in a cognitive category: an outset (i.e. the current state of the system) and a goal (i.e. the desired state). Cognitive systems transform the target system by means of generators and evaluators. Generators realize cognitive operations over a system by grouping morphisms, whilst evaluators group objects as a way to generalize outsets and goals to partially defined states. Meta-cognition emerges when the whole cognitive system is self-referenced as sub-states in the cognitive category, whilst learning must always be considered as a meta-cognitive process to maintain consistency. Several examples grounded in basic AI methods are provided as well., Comment: Laboratory for Intelligent Dynamics and Representation. Waseda University

The online algorithm design was proposed to handle the caching problem when the future information is unknown. And currently, it draws more and more attentions from the researchers from the areas of microgrid, where the production of renewables are unpredictable. In this note, we present a framework of randomized online algorithm design for the \textit{simple and tractable} problem. This framework hopes to provide a tractable design to design a randomized online algorithm, which can be proved to achieve the best competitive ratio by \textit{Yao's Principle}.

Most of computer science focuses on automatically solving given computational problems. I focus on automatically inventing or discovering problems in a way inspired by the playful behavior of animals and humans, to train a more and more general problem solver from scratch in an unsupervised fashion. Consider the infinite set of all computable descriptions of tasks with possibly computable solutions. The novel algorithmic framework POWERPLAY (2011) continually searches the space of possible pairs of new tasks and modifications of the current problem solver, until it finds a more powerful problem solver that provably solves all previously learned tasks plus the new one, while the unmodified predecessor does not. Wow-effects are achieved by continually making previously learned skills more efficient such that they require less time and space. New skills may (partially) re-use previously learned skills. POWERPLAY's search orders candidate pairs of tasks and solver modifications by their conditional computational (time & space) complexity, given the stored experience so far. The new task and its corresponding task-solving skill are those first found and validated. The computational costs of validating new tasks need not grow with task repertoire size. POWERPLAY's ongoing search for novelty keeps breaking the generalization abilities of its present solver. This is related to Goedel's sequence of increasingly powerful formal theories based on adding formerly unprovable statements to the axioms without affecting previously provable theorems. The continually increasing repertoire of problem solving procedures can be exploited by a parallel search for solutions to additional externally posed tasks. POWERPLAY may be viewed as a greedy but practical implementation of basic principles of creativity. A first experimental analysis can be found in separate papers [53,54]., Comment: 21 pages, additional connections to previous work, references to first experiments with POWERPLAY

In this paper we give a general geometrical framework for working with problems that can be described as a structure-preserving submersion defined on a suitable space with a geometrical structure. We give many examples of how to formulate familiar problems arsing from physics in our framework. Then, as an application of this framework we derive the new results of generalisations of the classical Herglotz-Noether theorem to arbitrary dimensions and to all conformally-flat ambient spaces. Possible further directions for research are also discussed.

The problem of estimating a generic phase-shift experienced by a quantum state is addressed for a generally degenerate phase shift operator. The optimal positive operator-valued measure is derived along with the optimal input state. Two relevant examples are analyzed: i) a multi-mode phase shift operator for multipath interferometry; ii) the two mode heterodyne phase detection., Comment: 11 pages. Elsart package used