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We investigate the typical sizes and shapes of sets of points obtained by irregularly tracking two-dimensional Brownian bridges. The tracking process consists of observing the path location at the arrival times of a non-homogeneous Poisson process on a finite time interval. The time varying intensity of this observation process is the tracking strategy. By analysing the gyration tensor of tracked points we prove two theorems which relate the tracking strategy to the average gyration radius, and to the asphericity -- a measure of how non-spherical the point set is. The act of tracking may be interpreted either as a process of observation, or as process of depositing time decaying "evidence" such as scent, environmental disturbance, or disease particles. We present examples of different strategies, and explore by simulation the effects of varying the total number of tracking points., Comment: 12 pages of the main article followed by the supplementary material

Optical activity and birefringence in the presence of multiple reflections for (100), (010) and (101) planes of quartz were studied by using a high-accuracy universal polarimeter (HAUP) in a temperature range between 20 and 300 °C. Likewise, optical activity in the presence of multiple reflections for the (001) plane of quartz was studied by using a conventional polarimeter in the same temperature range at 632.8 nm wavelength. In all cases a modulation of the HAUP parameters determined by linear least-square fit is observed when the temperature is changed. This effect can be explained by the multiple reflections within the slab of a high degree of plane parallelism. The Levenberg-Marquardt method of nonlinear least-square routines was used to compare the experimental values of the HAUP coefficients with the results obtained using the Jones formalism. The method gives directly the optimum temperature dependence of birefringence and gyration tensor components of quartz.

The complete optical-gyration tensor for a monoclinic crystal of has been determined by the tilter method at a wavelength of 680 nm. The tensor components in terms of rotatory power in relation to the principal axes of the indicatrix were found to be , with in the optic-axis directions. The absolute optical chirality has also been established by combining the structural chirality determination (through the x-ray anomalous scattering) and the signs of the gyration tensor components measured for the same crystal.

On the basis of the superposition principle for the optical anisotropy parameters and the Jones calculus methods a phenomenological model is developed to interpret the optical properties of absorbing, spatially modulated dielectric materials. The character of the normal electromagnetic waves propagated in a uniaxial, weakly dichroic crystal for which the complex gyration tensor is modulated with the simple square wave form is ascertained. The structure with a perfect modulation wave and the unipolar structure are considered and should describe the multidomain ferroelectric and incommensurately modulated phases. Specific crystal optical effects are found which have no analogues in uniform materials. Proceeding from the normal wave polarization the modulated crystal is shown to be equivalent to a low-symmetry biaxial crystal. The origin of a non-zero eigenwave ellipticity is discussed in connection with the problem of optical activity in macroscopically centrosymmetric incommensurate crystals of group.

This paper reports on the measurement of the optical properties of incommensurately modulated ((CH3)4N)2CuCl4 by using the High Accuracy Universal Polarimeter. With this technique it is possible to measure, simultaneously, linear birefringence, linear dichroism, circular birefringence, circular dichroism and the rotation of the optical indicatrix. Two different samples are used. The orientation of the first sample allows for the measurement of the gyration tensor element g33. In the other sample g13 can be measured. The optical effects are studied as a function of temperature, both in a region of zero linear dichroism and in a region of finite linear dichroism. The measurements reveal that the crystals have a low defect concentration. The presence of the incommensurate modulation is clearly revealed by the linear birefringence. The linear dichroism, on the other hand, appears to be unaffected by the modulation. It is interesting that if one averages the effect of the incommensurate modulation a crystal structure is obtained that is believed to be orthorhombic and centrosymmetric. Two effects are observed that are forbidden by the symmetry of this average structure. One of them is the rotation of the optical indicatrix, which contradicts an orthorhombic symmetry. The other is the onset of non-zero optical activity, approximately halfway through the incommensurate phase. In centrosymmetric media this effect is not allowed to occur. The observed temperature dependence of the optical activity g13 differs from the behaviour measured by other authors. It is discussed whether the observed effects can be attributed to a symmetry breaking by the incommensurate modulation.

The optical activity, birefringence and the indicatrix rotation of a (001) sample of (N(CH3)4)2ZnCl4 are measured in the high-temperature, incommensurate and partly lock-in phases with a new high-accuracy null polarimeter. The optical properties are shown to be sensitive to the annealing and X-irradiation of a sample via the defect concentration changes. The results can be explained by consideration of a modulated gyration tensor in the incommensurate phase, using the idea of modulation wave distortions imposed by defects.

Optical properties of LiNH4C4H4O6·H2O (LAT) were studied by means of a high-accuracy universal polarimeter (HAUP). The temperature dependences of the gyration tensor components were obtained. The elastogyration coefficients were determined. Rotation of the indicatrix takes place only around the b axis in the low-temperature phase. The rotation angle of the gyration surface was found to be three orders of magnitude larger than that of the indicatrix.

Optical gyration studies on Co3B7O13I and Cu3B7O13Cl were conducted by the high accuracy universal polarimeter (HAUP) method. Temperature dependences of the gyration tensor g 12 manifested sharp peaks at improper ferroelectric transition points. The electrogyration coefficient of Co3B7O13I also exhibited a sharp peak at the transition point. Taking our previous work of Fe3B7O13I into account, it was concluded that peculiar temperature dependences of gyration and the electrogyration coefficient are a common nature of ferroelectric boracite crystals.

An optical gyration study of Fe-I boracite (Fe3B7O13I) was first made by using the high accuracy universal poralimeter (HAUP) method. Temperature dependence of a gyration tensor component g 12 manifested a sharp peak at the improper ferroelectric phase transition point. We also studied electrogyration effects by applying electric field along the ferroelectric axis. Unusual electric field effects were revealed. For example, an electrogyration coefficient also exhibited a sharp peak at the transition point.

For pt.I see ibid., vol.19, p.4339 (1986). The theory of double sublattice antiferromagnets based on the equations dynamics, given in the first part of this paper, is developed. Normal coordinates of natural vibrations of the subsystem are plotted, when the sublattice spin value is not the integral of motion. It is shown that the almost flat motion of spins corresponds to the third (the lowest) frequency. The susceptibility to a high-frequency field chi ik( omega , T, H) is calculated and it is shown that the third branch of the spectrum should be indicated in the components of the gyration tensor, as well as in diagonal components of chi ik. The theory predicts the most quantitatively strong effects in the diagonal component chi ii perpendicular to the antiferromagnetism vector and to the external field. It is proved that damping conformed with the observed width of antiferromagnetic and paramagnetic resonance lines cannot overlap the effects of the third branch of the spectrum predicted by the theory. In conclusion the authors briefly discuss the possible indication of the third branch of the spectrum in low-temperature thermodynamics of the antiferromagnet.

The complicated structure of one of the phases of tetramethylammonium trichlorocuprate (TMACuCl3) is discussed. The observed deformations with respect to the high-symmetry space group have been decomposed into irreducible representations of this group (P63/mmc). The representations that induce the deformations are tau 5 (k=2 pi (0,0,1/5c)) and tau 6 (k=2 pi (0,0,2/5c)) (Kovalev notation). Structural resonance occurs between these two waves through anharmonic energy contributions. This resonance is such that in the resulting structure two building blocks are discernible, one block showing all characteristics of CsCuCl3 and one some of RbCuCl3. The elements of the gyration tensor have been calculated using single-crystal diffraction data. The chiral functions from which the tensor elements are calculated form fourth-order invariants when squared. This concept can be demonstrated on both the Jahn-Teller octahedra and the successive Cu and Cl layers of this compound. Furthermore, an explanation is given for the fact that some of the TMA groups are disordered, thus contributing to the free energy by an entropy term, and some are not. The superposition of the two waves is such that the E(X) in Jahn-Teller effect of the individual octahedra fits in precisely.

Structural (gyrotropic) phase transitions from optically inactive to optically active (gyrotropic) phases are analysed for nonmagnetic materials. The following basic results of group theoretical analysis are gathered for those equitranslational gyrotropic phase transitions which are associated with a single REP (irreducible or physically irreducible representations): e.g. the macroscopic transition parameter, the spontaneous components of the gyration tensor, the polarisation and deformation tensors newly onsetting at the transition, the indices of faintness, faint interactions, and switching forces. A simple characterisation of the domain structure of the ferroic phase is given in terms of these tensors.

This report describes a theoretical study of a convenient method for measuring the optical activity in crystals in which birefringence and optical activity coexist. The method was applied to right-handed quartz. The gyrations in the [100] direction and at 45° from the optical axis were measured. The gyration tensor component g 33 calculated from the gyrations measured in these two directions was found to be in good agreement with the value of g 33 measured by other workers in the direction of the c-axis (the optical axis) by conventional methods.