We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

From the symmetry point of view, micas may be classified as follows: those with all three octahedrally coordinated sites occupied by the same cation (homo-octahedral micas), those with only two of these sites occupied by the same cation (meso-octahedral micas), and those with the three sites occupied by different cations or by two different cations and a void, in an ordered manner (hetero-octahedral micas). For any of these three classes, mica polytypes, idealized in accordance with the generalized Pauling model, can be interpreted as OD structures consisting of octahedral OD layering and tetrahedral OD layering in which an interlayer cation plane is sandwiched between tetrahedral sheets. A mica layer built up by an octahedral sheet and two halves of tetrahedral sheets on either side consists of two OD packets linked by a two-fold rotation.

The orientation of any OD packet may be given by a number from 0 to 5 (related to a hexagonal coordinate system). A dot behind or before these numbers is used to denote the position of the octahedral layer (number + dot = orientational character). The displacement of a packet against its predecessor is characterized by a vector from the origin of a packet pn (or qn-1) to the origin of the adjacent packet pn+1 (or p2n). These displacements may also be symbolized by numbers from 0 to 5 (displacement characters); a zero displacement is symbolized by *. Any mica polytype (ordered or disordered) can thus be described by a two-line symbol. The orientational characters are located on the first line, and the displacement characters on the second. Any symbol, therefore denotes unequivocally the stacking layers in a polytype. The space-group symmetry of ordered polytypes follows directly from the symbol.

Single crystals of sudoite from Ottré, Belgium, allow confirmation of a dioctahedral 2:1 layer and a trioctahedral interlayer in a IIb arrangement. A regular 2-layer s structure is formed in which the octahedral stagger within both 2:1 layers is directed along X1 and adjacent layers are alternately displaced by a2/3 and a3/3. Poor quality crystals and twinning prevented three-dimensional refinement. One-dimensional refinement suggests that the smaller d(001) value of dioctahedral chlorites relative to trioctahedral species is due primarily to the thinner dioctahedral sheet.

Morphological symmetry abnormalities in cheliped appendages of alpheid shrimps are extremely rare and poorly recorded in the literature. A symmetric minor cheliped were, for example, observed in queen females belonging to Synalpheus eusocial species. Symmetric major chelipeds were now described in Synalpheus fritzmuelleri individuals living in shallow Brazilian waters. These individuals were found in symbiotic association with the bryozoan Schizoporella sp. (biogenic substrate) adhering to the pilings of Ubatuba Bay docks, São Paulo State. Only one of 20 sampled S. fritzmuelleri individuals presented anomalous symmetric chelipeds. Based on carapace length, size, and morphological features, the analyzed specimens seemed to be juvenile; thus, the hypothesis of anomalous condition can be directly linked to genetic inhibition of the mechanism accounting for major cheliped development in this ontogeny phase. Studies like the present one often provide remarkable information on animal morphology and can be used as reference in evolutionary assessments to be conducted in the future.

Hypertrophic ‘giant’ handaxes are a rare component of Acheulean assemblages, yet have been central to debates relating to the social, cognitive and cultural ‘meaning’ of these enigmatic tools. The authors examine giant handaxes from the perspective of the British record and suggest that they are chronologically patterned, with the great majority originating from contexts broadly associated with Marine Isotope Stage 9. Giant handaxes tend to have higher symmetry than non-giants, and extravagant forms, such as ficrons, are better represented; they may therefore be linked to incipient aesthetic sensibilities and, potentially, to changing cognition at the transition between the Lower and Middle Palaeolithic.

To determine the relationships between the symmetry of the overall pyrophyllite and talc structure and the symmetry of individual layers, the geometry and symmetry of each 2:1 layer of pyrophyllite and talc were analyzed. For each, the previously published, refined unit cell may be rotated clockwise by ~60° for comparison to a layer unit cell. In pyrophyllite, the layer unit cell is ideal and shown to be orthogonal with C2/m symmetry. The agreement between the refined atomic coordinates and those calculated for the layer with C2/m symmetry confirms that the symmetry of the pyrophyllite layer is C2/m. The obliquity of the pyrophyllite refined cell results from the layer stacking and the choice of unit cell, but the interlayer stacking sequence does not disturb the layer symmetry. In contrast, talc has an oblique layer cell, without a mirror plane. For the most part, the distortion of the talc 2:1 layer is probably caused by an elongation of unshared O-O lateral edges around M1 that creates a slight corrugation of the octahedral sheet surface. Perhaps of lesser importance, the distortion of the talc layer cell may result from Coulombic interactions between cations of adjacent layers, and these cation-to-cation distances are sufficiently large (~6–7.5 Å) that the weak van der Waals forces that stabilize the stacking are not overcome. Because pyrophyllite has a vacant octahedral site, similar interactions are not present, and this results in a more idealized layer symmetry.

Phyllosilicates consisting of layers with an orthogonal cell and mirror plane (pyrophyllite, kaolinite, sudoite) were shown to have similar stacking faults. In these structures, the 2:1 or 1:1 layers have uniform orientation, and stacking faults occur owing to interstratifications of two alternative interlayer displacements in the same crystal that are related by a mirror plane in the projection on the (001) plane. In talc, stacking faults are associated with layer rotations by ±120°, whereas the lateral displacement between the adjacent tetrahedral sheets across the interlayer region is relatively ordered.

This study aimed to determine the load-bearing capacity of trotting Yonaguni ponies using gait analysis. The Yonaguni pony is one of the Japanese landrace horses, and has normal gait characteristics when trotting. As a breed they are small in stature and hence susceptible to the effect of the rider's weight. It is therefore important to determine their load-bearing capacity as regards to their welfare. Ten Yonaguni ponies with a (mean ± SD) height at withers of 122 (± 2.9) cm had a marker attached to their chests, and their unridden gait was recorded using two high-resolution hybrid cameras while they trotted along a short (40 m), straight course. In total, nine tests were performed for each horse: the first with a 0-kg load; seven with randomly loaded weights weighing 10-70 kg; and then a final test again with a 0-kg load. Three-dimensional movement of the marker was analysed using a motion capture system. The time series of vertical displacement of the marker underwent spectrum analysis, and the autocorrelation coefficient was calculated. The first two peaks of the autocorrelation were defined as symmetry and gait regularity, and their sum was defined as stability. Symmetry (no unit) in the 70-kg test (0.53) was lower than that in the first 0-kg test (0.68), and stability (no unit) in the 70-kg test (1.16) was lower than that in the first 0-kg test (1.41). We concluded that the maximum permissible load for a trotting Yonaguni pony is < 70 kg, which represents 33% of its bodyweight. To promote welfare, it is important to determine the load-bearing capacity for individual types of horse.

Symmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the range of a symmetry action; a generalised Noether (pre-symplectic) operator constructed from any non-variational adjoint-symmetry. These results are illustrated here by considering five examples of physically interesting nonlinear PDE systems – nonlinear reaction-diffusion equations, Navier-Stokes equations for compressible viscous fluid flow, surface-gravity water wave equations, coupled solitary wave equations and a nonlinear acoustic equation.

Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearisation (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra. Solutions of the adjoint linearisation equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether’s theorem, which has a modern generalisation to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-symplectic (Noether) operator, which leads to the construction of symplectic 2-form and Poisson bracket for evolution systems. The generalised KdV equation in potential form is used to illustrate all of the results.