Nonlinear superposition rule for Abel's equation

A nonlinear superposition rule for Abel's cubic differential equation involving four particular solutions can be replaced by a superposition rule involving only two particular solutions. Chiellini's restrictions on the particular solutions are found to be satisfied.     

Real-time spectral imaging in three spatial dimensions

We report what is to our knowledge the first volume-holographic optical imaging instrument with the capability to return three-dimensional spatial as well as spectral information about semitranslucent microscopic objects in a single measurement. The four-dimensional volume-holographic microscope is characterized theoretically and experimentally by use of fluorescent microspheres as objects.     

Invariant solutions of Liouville's equation in Robertson-Walker space-times

We find all solutions of Liouville's equation in Robertson-Walker space-times that are either spatially homogeneous or isotropic or both. Some of these solutions depend on constants of motion that are not generated by Killing vectors. We indicate how these solutions may be used to find Einstein-Liouville solutions.     

On ellipsoidal solutions of Liouville's equation in general relativity

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.     

General solution of Liouville's equation in Robertson-Walker space-times

We obtain the general solution of Liouville's equation in Robertson-Walker space-times, based on Killing vector constants of the motion and a new time-dependent first integral. For photons the solution is generated by conformai Killing vectors.     

Invariant endomorphism and conserved functionals of Liouville's equation

An invariant tensor field which accounts for symmetries and conserved functionals of Liouville's equation, is constructed. A simple connection with the Korteweg-de Vries equation is pointed out.     

EPR oximetry in three spatial dimensions using sparse spin distribution

A method is presented to use continuous wave electron paramagnetic resonance imaging for rapid measurement of oxygen partial pressure in three spatial dimensions. A particulate paramagnetic probe is employed to create a sparse distribution of spins in a volume of interest. Information encoding location and spectral linewidth is collected by varying the spatial orientation and strength of an applied magnetic gradient field. Data processing exploits the spatial sparseness of spins to detect voxels with nonzero spin and to estimate the spectral linewidth for those voxels. The parsimonious representation of spin locations and linewidths permits an order of magnitude reduction in data acquisition time, compared to four-dimensional tomographic reconstruction using traditional spectral-spatial imaging. The proposed oximetry method is experimentally demonstrated for a lithium octa-n-butoxy naphthalocyanine (LiNc–BuO) probe using an L-band EPR spectrometer.     

Painlevé analysis for a nonlinear schrödinger equation in three dimensions

A Painlevé analysis is performed for the nonlinear Schrödinger equation in (2+1) dimensions following the methodology of Weiss et al. simplified in the sense of Kruskal. At least for one branch it is found that the required number of arbitrary functions (as demanded by the Cauchy-Kovalevskaya theorem) exists, signalling complete integrability.     

Nonlinear superposition law for generalized Ermakov systems

We discuss a generalized Ermakov system for which a general nonlinear superposition law exists. We give an explicit example of the use of the new superposition law.     

On the recursion operator for the KP hierarchy in two and three spatial dimensions

A new recursion operator generating the KP hierarchy is found. Much simpler than the one previously known, it possesses all the good canonical properties and generates the elementary Bäcklund transformation. This operator naturally suggests a canonical recursion operator in 3+1 dimensions generating a bihamiltonian hierarchy of nonlinear evolution equations, their Bäcklund transformation and nonlinear superposition principle.     

Evolution of the initial box-signal for time-fractional diffusion-wave equation in a case of different spatial dimensions

In the case of time-fractional diffusion-wave equation considered in the spatial domain −∞<x<∞, evolution of initial box-signal was investigated by Mainardi [F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals 7 (1996) 1461-1477]. In the present paper, we supplement Mainardi’s results with additional numerical calculations illustrating the behavior of the solution and solve the corresponding problems for axisymmetric and central symmetric cases. The obtained results show an unusual behavior of solutions.     

A method of solution of the ornstein-zernike equation in one and three dimensions

An elegant analytic method is presented to derive a modified form of the Ornstein-Zernike equation which not only solves the statistical mechanical problem for hard rods and spheres but can be used to find solutions for arbitrary non-vanishing direct correlation functions.     

Analysis of spatial relationships in three dimensions: tools for the study of nerve cell patterning

Background

Multiple technologies have been brought to bear on understanding the three-dimensional morphology of individual neurons and glia within the brain, but little progress has been made on understanding the rules controlling cellular patterning. We describe new matlab-based software tools, now available to the scientific community, permitting the calculation of spatial statistics associated with 3D point patterns. The analyses are largely derived from the Delaunay tessellation of the field, including the nearest neighbor and Voronoi domain analyses, and from the spatial autocorrelogram.

Results

Our tools enable the analysis of the spatial relationship between neurons within the central nervous system in 3D, and permit the modeling of these fields based on lattice-like simulations, and on simulations of minimal-distance spacing rules. Here we demonstrate the utility of our analysis methods to discriminate between two different simulated neuronal populations.

Conclusion

Together, these tools can be used to reveal the presence of nerve cell patterning and to model its foundation, in turn informing on the potential developmental mechanisms that govern its establishment. Furthermore, in conjunction with analyses of dendritic morphology, they can be used to determine the degree of dendritic coverage within a volume of tissue exhibited by mature nerve cells.     

Exploding dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one and two spatial dimensions

We review the work on exploding dissipative solitons in one and two spatial dimensions. Features covered include: the transition from modulated to exploding dissipative solitons, the analogue of the Ruelle-Takens scenario for dissipative solitons, inducing exploding dissipative solitons by noise, two classes of exploding dissipative solitons in two spatial dimensions, diffusing asymmetric exploding dissipative solitons as a model for a two-dimensional extended chaotic system. As a perspective we outline the interaction of exploding dissipative solitons with quasi one-dimensional dissipative solitons, breathing quasi one-dimensional solutions and their possible connection with experimental results on convection, and the occurence of exploding dissipative solitons in reaction-diffusion systems. It is a great pleasure to dedicate this work to our long-time friend Hans (Prof. Dr. Hans Jürgen Herrmann) on the occasion of his 60th birthday.     

Nonlinear superposition law and Feynman propagator

We demonstrate the usefulness of the nonlinear superposition law of Ray and Reid in connection with the Feynman propagator. We derive the propagator for the problem of a time-dependent variable mass particle subject to a harmonic potential with a time-dependent frequency by taking direct recourse to the corresponding Schrödinger equation.     

Diffraction in three dimensions

Atmospheric Convection . By K. A. Emanuel. 1994, £47.50 (hbk), pp. x+580. Oxford, ISBN 0 19 506630 8. Scope: text. Level: postgraduate.

Automotive Sensors . By M. H. Westbrook and J. D. Turner. 1994, £75.00, US$149.00 (hbk), pp. xviii + 253. IOP, ISBN 0 7903 0293 3. Scope: monograph. Level: specialist.

Computation in Modem Physics . By W. R. Gibbs. 1994, £38.00 (hbk), pp. xvii + 348. World Scientific, ISBN 981 02 2044 8. Scope: text. Level: undergraduate and postgraduate.

Electron Correlations in Molecules and Solids , 2nd Edn. By P. Fulde. 1993, DM78.00 (pbk), pp. xiv + 422. Springer-Verlag, ISBN 3 540 56376 8. Scope: monograph. Level: postgraduate and specialist.

Field Theory, A Path Integral Approach . By A. Das. 1994, £48.00 (hbk), pp. xv + 399. World Scientific, ISBN 981 02 1396 4. Scope: lecture notes. Level: postgraduate.

Infrared Technology Fundamentals , 2nd Edn, revised and expanded. By M. Schlessinger. 1994, US$125.00 (hbk), pp. xii + 462. Dekker, ISBN 0 8247 9259 9. Scope: monograph. Level: postgraduate and specialist.

Oceanography and Acoustics . Edited by A. R. Robinson and D. Lee. 1994, US$65.00 (hbk), pp. xvii + 257. AIP, ISBN 1 56396 203 9. Scope: multi-author review. Level: postgraduate and specialist.

Optoelectronic Integration: Physics, Technology and Applications . Edited by O. Wada. 1994, Dfl. 275.00, US$130.00, £96.25 (hbk), pp. x + 458. Kluwer, ISBN 0 7923 9453 4. Scope: multi-author review. Level: postgraduate and specialist.

Photonic Devices and Systems . Edited by R. C. Hunsperger. 1994, US$135.00 (hbk), pp. xi + 426. Dekker, ISBN 0 8247 9243 2. Scope: multi-author review. Level: postgraduate and specialist.

Physics and Our View of the World . Edited by J. Hilgevoord. 1994, US$22.95 (pbk), pp. 304. Cambridge, ISBN 0 521 47680 1. Scope: multi-author review. Level: scientists and philosophers of science.

Polymer Physics . By U.W. Gedde. 1995, £49.00 (hbk), £24.99 (pbk), pp. x + 298. Chapman &; Hall, ISBN 0 412 62640. Scope: textbook. Level: undergraduate and postgraduate.

Quantization of Gauge Systems . By M. Henneaux and C. Teitelboim. 1994, US$29.95, £25.00 (pbk) pp. xxii + 520. Princeton, ISBN 0 691 03769 8. Scope: textbook. Level: postgraduate and specialist.

Semiconductor Raman Lasers . By K. Suto and J.-I. Nlshlzawa. 1994, £63.00 (hbk), pp. ix + 226. Artech, ISBN 0 89006 667 1. Scope: monograph. Level: specialist.

Shapes and Shells in Nuclear Structure . By S. G. Nilsson and I. Ragnarsson. 1995, £60.00, US$89.95 (hbk), pp. xiv + 408. Cambridge, ISBN 0 521 37377 8. Scope: textbook. Level: postgraduate.

Supersymmetric Gauge Field Theory and String Theory . By D. Bailin and A. Love. 1994, £80.00, US$160.00 (hbk), £27.50, US$49.00 (pbk), pp. xii + 332. IOP, ISBN 0 7503 0268 2. Scope: textbook. Level: postgraduate.

Short notices Advanced Series in Neuroscience , Vol. 3. Biology and Computation, A Physicist's Choice. Reprint volume. Edited by H. Gutfreund and G. Toulouse. 1994, £94.00 (hbk) pp. xv + 852. World Scientific, ISBN 981 02 1405 7. Scope: reprints. Level: postgraduate and specialist.

Electronic Structure of Refractory Carbides and Nitrides . Edited by V. A. Gubanov, A.L. Ivanovsky and V. P. Zhukov. 1994, $79.95 (hbk) pp. xiv + 256. Cambridge University Press, ISBN 0 521 41885 2. Scope: monograph. Level: postgraduate and specialist.

Fritz London: A Scientific Biography . By K. Gavroglu. 1995, £50.00 (hbk), pp. xxiv + 299. Cambridge University Press, ISBN 0 521 43273 1. Scope: biography. Level: general reader.

Gallium Arsenide and Related Compounds 1993 . Edited by H. S. Rupprecht and G. Weinmann. 1994, £115.00, US$230.00 (hbk), pp. xxiv + 842. IOP, ISBN 0 7503 0295 X. Scope: conference proceedings. Level: postgraduate and specialist.

Matvei Petrovich Bronstein and Soviet Physics in the Thirties . By G. E. Gorelik and V. Y. Frenkel. 1994, £65.00, DM 168.00. sFr. 148.00 (hbk), pp. 208. Birkhauser, ISBN 3 7643 2752 9. Scope: biography. Level: non-specialist.

Physical Chemistry . 5th edn. By P. W. Atkins. 1994, £24.00 (pbk), pp. xix + 1145. Oxford, ISBN 0 19 855730 2. Scope textbook. Level: undergraduate.

Polyominoes. Puzzles, Patterns, Problems, and Packings . Revised and expanded 2nd edn. By S.W. Golomb. 1994, US$24.95, £19.95 (hbk) pp. xii + 181. Princeton University Press, ISBN 0 691 08573 0. Scope: text. Level: non-specialist.

Supercollider 5 . Edited by P. Hale. 1994, US$195.00 (hbk), pp. xxviii + 970. Plenum, ISBN 0 306 44705 3. Scope: conference proceedings. Level: postgraduate and specialist.

World Scientific Series in 20th Century Physics , Vol. 5. Selected Papers of Abdus Salam (With Commentary). Edited by A. Ali, C. Isham, T. Kibble and Riazuddin. 1994, £64.00 (hbk) pp. xiv + 679. World Scientific, ISBN 981 02 1662 9. Scope: reprinted papers. Level: specialist.     

Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions

A Fredholm second kind integral equation (SKIE) formulation is constructed for the Dirichlet problem of the biharmonic equation in three dimensions. A fast numerical algorithm is developed based on the constructed SKIE. Its performance is illustrated via several numerical examples.     

A nonlinear superposition rule for solutions of the Milne-Pinney equation

A superposition rule for two solutions of a Milne-Pinney equation is derived.