A Note on Maximum Shear

The problem of the determination at any point P in a body of that pair of infinitesimal material line elements which suffers the maximum shear in a deformation has been solved [1]. Here that problem is revisited and a short proof, of geometrical type, of the result is presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

A new statistical method for the automated detection of peaks in UV-DAD chromatograms of a sample mixture

One of the major issues within the context of the fully automated development of chromatographic methods consists of the automated detection and identification of peaks coming from complex samples such as multi-component pharmaceutical formulations or stability studies of these formulations. The same problem can also occur with plant materials or biological matrices. This step is thus critical and time-consuming, especially when a Design of Experiments (DOE) approach is used to generate chromatograms. The use of DOE will often maximize the changes of the analytical conditions in order to explore an experimental domain. Unfortunately, this generally provides very different and “unpredictable” chromatograms which can be difficult to interpret, thus complicating peak detection and peak tracking (i.e. matching peaks among all the chromatograms). In this context, Independent Components Analysis (ICA), a new statistically based signal processing methods was investigated to solve this problem. The ICA principle assumes that the observed signal is the resultant of several phenomena (known as sources) and that all these sources are statistically independent. Under those assumptions, ICA is able to recover the sources which will have a high probability of representing the constitutive components of a chromatogram. In the present study, ICA was successfully applied for the first time to HPLC–UV-DAD chromatograms and it was shown that ICA allows differentiation of noise and artifact components from those of interest by applying clustering methods based on high-order statistics computed on these components. Furthermore, on the basis of the described numerical strategy, it was also possible to reconstruct a cleaned chromatogram with minimum influence of noise and baseline artifacts. This can present a significant advance towards the objective of providing helpful tools for the automated development of liquid chromatography (LC) methods. It seems that analytical investigations could be shortened when using this type of methodologies.     

Thermomechanics of the interface between a body and its environment

We formulate integral statements of force balance, energy balance, and entropy imbalance for an interface between a body and its environment. These statements account for interfacial energy, entropy, and stress but neglect the inertia of the interface. Our final results consist of boundary conditions describing thermomechanical interactions between the body and its environment. In their most general forms, these results are partial differential equations that account for dissipation and encompass as special cases Navier’s slip law, Newton’s law of cooling, and Kirchhoff’s law of radiation. When dissipation is neglected, our results reduce to the well-known zero-slip, free-surface, zero-shear, prescribed temperature, and flux-free conditions. Dedicated to James K. Knowles: teacher, colleague, friend     

In General There Are No Unsheared Tetrads

Suppose the principal stretches are all different at a point P in a deformed body. In this case, it has been shown [1] that generally there is an infinity of non coplanar infinitesimal material line elements at P which remain unsheared following the deformation – that is, the angle between the arms of each pair of material line elements forming the triad remains unchanged. Here it is shown that in this case when all three principal stretches at P are different, there is no set of four infinitesimal material line elements, no three of which are coplanar, and such that the angle between each pair of the six pairs of material line elements is unchanged following the deformation. It is only when all three principal stretches at P are equal to each other, that there are unsheared tetrads at P, and in that case all tetrads are unsheared. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

The Galois group of a radical extension of the rationals

The Galois group of the splitting field of an irreducible binomialx ^2e −a overQ is computed explicitly as a full subgroup of the holomorph of the cyclic group of order 2 ^e . The general casex ⁿ −a is also effectively computed.     

Spin-lattice coupling coefficients of Mn2+ ions in stacking faults of ZnS

The spin-lattice coupling coefficients (SLCC) of Mn²⁺ ions in axial sites PN of cubic ZnS crystals containing stacking faults have been measured by an uniaxial stress method. These coefficients were correctly interpreted in an ionic model by taking into account the Blume-Orbach mechanism adapted to a C_3v symmetry and a relativistic mechanism whose importance has been previously demonstrated. Finally, a comparison between the experimental SLCC's for Mn₂₊ in the cubic sites and two axial sites PN and AS due to stacking faults shows that they are roughly identical as predicted theoretically.