Research on the reasoning, teaching and learning of probability and uncertainty

In this editorial, we set out the aims in the call to publish papers on informal statistical inference, randomness, modelling and risk. We discuss how the papers published in this issue have responded to those aims. In particular, we note how the nine papers contribute to some of the major debates in mathematics and statistics education, often taking contrasting positions. Such debates range across: (1) whether knowledge is fractured or takes the form of mental models; (2) heuristic or intuitive thinking versus operational thinking as for example in dual process theory; (3) the role of different epistemic resources, such as perceptions, modelling, imagery, in the development of probabilistic reasoning; (4) how design and situation impact upon probabilistic learning.     

Fuzzy qualitative trigonometry

This paper presents a fuzzy qualitative representation of conventional trigonometry with the goal of bridging the gap between symbolic cognitive functions and numerical sensing & control tasks in the domain of physical systems, especially in intelligent robotics. Fuzzy qualitative coordinates are defined by replacing a unit circle with a fuzzy qualitative circle; a Cartesian translation and orientation are defined by their normalized fuzzy partitions. Conventional trigonometric functions, rules and the extensions to triangles in Euclidean space are converted into their counterparts in fuzzy qualitative coordinates using fuzzy logic and qualitative reasoning techniques. This approach provides a promising representation transformation interface to analyze general trigonometry-related physical systems from an artificial intelligence perspective.Fuzzy qualitative trigonometry has been implemented as a MATLAB toolbox named XTRIG in terms of 4-tuple fuzzy numbers. Examples are given throughout the paper to demonstrate the characteristics of fuzzy qualitative trigonometry. One of the examples focuses on robot kinematics and also explains how contributions could be made by fuzzy qualitative trigonometry to the intelligent connection of low-level sensing & control tasks to high-level cognitive tasks.     

Topologically knotted Lagrangians in simply connected four-manifolds

Vidussi was the first to construct knotted Lagrangian tori in simply connected four-dimensional manifolds. Fintushel and Stern introduced a second way to detect such knotting. This note demonstrates that similar examples may be distinguished by the fundamental group of the exterior.

     

Nuclear energy conversion with stacks of graphene nanocapacitors

The efficiency of conventional techniques used to harvest energy in nuclear reactors lies around 35%. This limit exists, because the nuclear energy is converted to electrical energy via heat engines. We study an alternative approach where the kinetic energy of nuclear reaction products is directly converted into electric energy in a stack of charged capacitors with a gap size of 500 nm and graphene electrodes. Graphene is expected to be chemically and mechanically stable in high‐radiation environments, because its tensile strength of 130 GPa is very large, about 100 times larger than most metals. The dielectric strength of such nanocapacitors exceeds 1 GV/m, because avalanching is suppressed at small gap sizes. In a 1 GV/m electric field charged nuclear reaction products, such as 5.6 MeV alpha particles, come to rest in of a stack with 5000 nanocapacitors. We show that during the deceleration process more than 90% of kinetic energy of charged nuclear reaction products is converted to electric energy and stored as electric energy in the stack. Each stack is 2.5‐mm thick and produces a high‐voltage DC current. A device with a 1‐Ci²⁴¹Am source is expected to generate 22 mW of electric power.     

Cartan-decomposition subgroups of

For and , we give explicit, practical conditions that determine whether or not a closed, connected subgroup of has the property that there exists a compact subset of with . To do this, we fix a Cartan decomposition of , and then carry out an approximate calculation of for each closed, connected subgroup of .

     

Minimal regular polygons serving as universal covers in R 2

A universal cover is a set K with the property that each set of unit diameter is a subset of a congruent copy of K. It is shown that the smallest regular n-gon, for fixed n 4, which serves as an universal cover in R ² is the smallest regular n-gon covering a Reuleaux triangle of unit width.     

Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

Opportunities and challenges in using WPAN and WLAN technologies in medical environments [Accepted from Open Call]

The present view of medical environments, where isolated networks are used for IT and medical applications, is changing toward an integrated heterogeneous network scenario that can support a wide range of applications. WPAN and WLAN technologies play a fundamental role in enabling such integrated environment that is expected to support both medical and nonmedical applications. The ultimate goal is to exploit WPAN and WLAN technologies, as well as other wireless networks, such as 3G cellular systems and satellite networks, to support highly efficient medical care delivery, anytime and anywhere. However, the life-critical nature of some medical applications imposes additional challenges that have not been considered in nonmedical scenarios. This article discusses some future scenarios where WLAN and WPAN technologies can be used to provide an integrated and ubiquitous network in medical environments, and identifies the main issues to be addressed in order to meet the QoS requirements of different medical applications when operating in integrated environment     

Environmentally benign nanomixing by sonication in high-pressure carbon dioxide

Due to the increased use of nanocomposites, mixing at nanoscale has become important. Current mixing techniques can be classified into: (a) dry mixing (mechanical mixing), (b) wet mixing, and (c) simultaneous production of mixed nanoparticles (when possible). Dry mixing is in general not effective in achieving desired mixing at nanoscale, whereas wet mixing suffers from different disadvantages like nanomaterial of interest should be insoluble, has to wet the liquid, and involves additional steps of filtration and drying. This paper examines the use of pressurized carbon dioxide having high density and low viscosity to replace the liquids (e.g., n-hexane, toluene). Ultrasound is applied to the suspension of nanopowders in gaseous and supercritical carbon dioxide where high impact collisions during sonication help mixing and the final mixture is obtained by simple depressurization. The method is tested for binary mixture of alumina/silica, silica/titania, MWNT (multiwalled carbon nanotubes)/silica, and MWNT/titania. The effects of sonication intensity and pressure on the degree of mixing are studied. Comparative study is also done with liquid n-hexane as a mixing media. Quantitative characterization (e.g., mean composition standard deviation, intensity of segregation) of mixing of alumina/silica and silica/titania is done with energy-dispersive X-ray spectroscopy, and that of MWNT/silica and MWNT/titania is done using field-emission scanning electron microscopy and day-light illumination spectrophotometry. Results show that mixing in carbon dioxide at higher ultrasound amplitudes is as good as in liquid n-hexane, and the final mixed product does not contain any residual media as in the case of liquid n-hexane.