The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares

Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of symmetry are studied, including classes that are generalizations of those categorized by Dudeney in 1917. We show that two classes of such magic squares are singular and the eigenspectra of such magic squares are highly structured. Lastly, we prove that natural magic squares of singly-even order of these classes do note exist.Comment: 26 page

Real Time Detection and Analysis of Facial Features to Measure Student Engagement with Learning Objects

This paper describes a software application that records student engagement in an on-screen task. The application records in real time the on-screen activity and simultaneously estimates the emotional state and head pose of the learner. The head pose is used to detect when the screen is being viewed and the emotional state provides feedback on the form of engagement. The application works without recording images of the learner. On completing the task, the percentage of time spent viewing the screen and statistics on emotional state (neutral, happy, sad) are produced. A graph depicting the learner’s engagement and emotional state synchronised with the screen captured video is also produced. It is envisaged that the tool will find application in learning activity and learning object design

Augmented saddle point formulation of the steady-state Stefan--Maxwell diffusion problem

We investigate structure-preserving finite element discretizations of the steady-state Stefan--Maxwell diffusion problem which governs diffusion within a phase consisting of multiple species. An approach inspired by augmented Lagrangian methods allows us to construct a symmetric positive definite augmented Onsager transport matrix, which in turn leads to an effective numerical algorithm. We prove inf-sup conditions for the continuous and discrete linearized systems and obtain error estimates for a phase consisting of an arbitrary number of species. The discretization preserves the thermodynamically fundamental Gibbs--Duhem equation to machine precision independent of mesh size. The results are illustrated with numerical examples, including an application to modelling the diffusion of oxygen, carbon dioxide, water vapour and nitrogen in the lungs.Comment: 27 pages, 5 figure

Structural electroneutrality in Onsager-Stefan-Maxwell transport with charged species

We present a method to embed local electroneutrality within Onsager-Stefan-Maxwell electrolytic-transport models, circumventing their formulation as differential systems with an algebraic constraint. Flux-explicit transport laws are formulated for general multicomponent electrolytes, in which the conductivity, component diffusivities, and transference numbers relate to Stefan-Maxwell coefficients through invertible matrix calculations. A construction we call a `salt-charge basis' implements Guggenheim's transformation of species electrochemical potentials into combinations describing a minimal set of neutral components, leaving a unique combination associated with electricity. Defining conjugate component concentrations and fluxes that preserve the structures of the Gibbs function and energy dissipation retains symmetric Onsager reciprocal relations. The framework reproduces Newman's constitutive laws for binary electrolytes and the Pollard-Newman laws for molten salts; we also propose laws for salt solutions in two-solvent blends, such as lithium-ion-battery electrolytes. Finally, we simulate a potentiostatic Hull cell containing a non-ideal binary electrolyte with concentration-dependent properties

Finite element methods for multicomponent convection-diffusion

We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons

Decade time-scale modulation of low mass X-ray binaries

Regular observations by the All Sky Monitor aboard the Rossi X-ray Timing Explorer satellite have yielded well-sampled light-curves with a time baseline of over ten years. We find that up to eight of the sixteen brightest persistent low mass X-ray binaries show significant, possible sinusoidal, variations with periods of order ten years. We speculate on its possible origin and prevalence in the population of low mass X-ray binaries and we find the presence of a third object in the system, or long-period variability intrinsic to the donor star, as being attractive origins for the X-ray flux modulation we detect. For some of the objects in which we do not detect a signal, there is substantial short-term variation which may hide modest modulation on long time-scales. Decade time-scale modulations may thus be even more common.Comment: 8 pages, 4 figures, 2 tables. Accepted by MNRA

Finite element methods for multicomponent convection-diffusion

We develop finite element methods for coupling the steady-state Onsager--Stefan--Maxwell equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization, and extensibility to the transient, anisothermal, and non-ideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the mass-average constraint in the Onsager--Stefan--Maxwell equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting the structure of linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to non-ideal fluids with a simulation of the microfluidic mixing of hydrocarbons

Correlated Optical/X-ray Long-term Variability in LMXB 4U1636-536

We have conducted a 3-month program of simultaneous optical, soft and hard X-ray monitoring of the LMXB 4U1636-536/V801 Ara using the SMARTS 1.3m telescope and archival RXTE/ASM and Swift/XRT data. 4U1636-536 has been exhibiting a large amplitude, quasi-periodic variability since 2002 when its X-ray flux dramatically declined by roughly an order of magnitude. We confirmed that the anti-correlation between soft (2-12 keV) and hard (> 20 keV) X-rays, first investigated by Shih et al. (2005), is not an isolated event but a fundamental characteristic of this source's variability properties. However, the variability itself is neither strictly stable nor changing on an even longer characteristic timescale. We also demonstrate that the optical counterpart varies on the same timescale, and is correlated with the soft, and not the hard, X-rays. This clearly shows that X-ray reprocessing in LMXB discs is mainly driven by soft X-rays. The X-ray spectra in different epochs of the variability revealed a change of spectral characteristics which resemble the state change of black hole X-ray binaries. All the evidence suggests that 4U1636-536 is frequently (~monthly) undergoing X-ray state transitions, a characteristic feature of X-ray novae with their wide range of luminosities associated with outburst events. In its current behavioural mode, this makes 4U1636-536 an ideal target for investigating the details of state changes in luminous X-ray binaries.Comment: 7 pages, 6 figures, accepted for publication in MNRA