Paul Erdös

Theory of the locomotion of nematodes: Dynamics of undulatory progression on a surface.

We develop a model of the undulatory locomotion of nematodes, in particular that of Caenorhabditis elegans, based on mechanics. The model takes into account the most important forces acting on a moving worm and allows the computer simulation of a creeping nematode. These forces are produced by the interior pressure in the liquid-filled body cavity, the elasticity of the cuticle, the excitation of certain sets of muscles and the friction between the body and its support.We propose that muscle excitation patterns can be generated by stretch receptor control. By solving numerically the equations of motion of the model of the nematode, we demonstrate that these muscle excitation patterns are suitable for the propulsion of the animal.

Magnetic susceptibility and the phase transition of NpO2

The magnetic susceptibility and the induced magnetic moment of NpO2 single crystals were measured between 3 and 300 K and in fields up to 1.5 T. The resistivity was measured above 150 K on pressed powder pellets. A band gap of 0.4 eV is found. Three theoretical models are discussed to explain the nature of the phase transition at T0 = 25.4 K and the absence of Np magnetic moment below T0. The crystal field model shows that the Γ(2)8 quartet of 5f3, split by quadrupolar interaction- induced distortion which sets in below T0, may have a quenched moment at the theoretically expected value x = −0.74 of the crystal field parameter. Other models based on the Np3+(5f4) configuration explain the transition either as an antiferromagnetic ordering of s = 12 holes located in that half of the oxygen cubes which contain no Np in the fluorite structure, or as an order-disorder transformation of O- and O2− ions. The calculated susceptibility is in fair agreement with the measurement. Further experiments are proposed to test the models.

Floating equilibrium of symmetrical objects and the breaking of symmetry. Part 1: Prisms

The equilibrium configurations of solid prisms of square and equilateral triangular cross section floating in a liquid are examined. It is found that these bodies float in different symmetrical or asymmetrical positions with respect to the vertical plane depending on the solid–liquid specific mass ratio, or depending on the height at which the body’s axis is held above the liquid level.

The neural basis of the locomotion of nematodes

A model of electrotonic neurons is presented which allows computer simulation of a physiologically realistic neural network, such as that found in nematodes. The undulatory locomotion of Caenorhabditis elegans is investigated by solving the equations of motion of a segmented model of the discretized body taking into account all internal and external forces. The spatio-temporal muscle excitation patterns which produce locomotion are determined, and it is concluded that these are most probably generated by stretch receptor cells and signals which globally turn on or off the neural circuitry governing forward or backward motion. The results are illustrated in the form of a computer generated videotape and compared with the observed motion of a nematode.

Floating equilibrium of symmetrical objects and the breaking of symmetry. Part 2: The cube, the octahedron, and the tetrahedron

The analysis of the equilibrium configurations of floating bodies (see part 1) is extended to the cube, for which a complete discussion is given. The symmetric floating configurations of the octahedron and of the tetrahedron are also examined.

Resistance of a one-dimensional periodic chain of random potentials

Recent results published by a number of authors concerning transmission properties of one-dimensional disordered systems of potentials are shown to be special cases of the general theory for these systems published in 1982. We also show that some of the restrictive assumptions used by these authors are unnecessary. Specific errors in these papers have been noted and corrected. In addition we have presented some new results concerning the convergence of the ensemble average resistance of these one-dimensional systems for some particular cases at resonant and mid-zone energies. It is shown that for certain cases the standard deviation of the resistance, which is calculated analytically, converges as the chain increases in length.

Spiraling the Earth with C. G. J. Jacobi

The simple requirement that one should move on the surface of a sphere with constant speed while maintaining a constant angular velocity with respect to a fixed diameter, leads to a path whose cylindrical coordinates turn out to be given by the Jacobian elliptic functions. Many properties of these functions can be derived and visualized using this path, known as Seiffert’s spiral.

Magnetic properties of UBr 3

The magnetic behavior of UBr/sub 3/ is explained by an anisotropic exchange molecular field model. The two magnetic phase transitions at T/sub N/ = 5.3 K and T/sub t/ = 3 K are obtained as a result of mini- mizing the free energy with respect to the relative occupation probabilities of crystal-field levels. Below T/sub t/ all U-ion moments are ordered; for T/sub t/

Currents in a superconducting loop with a branch connected to a current-carrying infinite wire

Using the full nonlinear Ginzburg-Landau equations, we study the behaviour of a particular superconducting micronetwork near its transition temperature: a circular loop connected to an infinitely long wire by a branch of length L. In this mesoscopic structure the superconducting condensate is confined by the boundary conditions, leading to nonlocal current effects. When a current iw is flowing in the infinitely long wire, the current i? in the loop can be found by numerical methods. This allows us to determine the relationship between the critical current ic in the loop, at which superconductivity disappears, and the current iw. When iw is zero or small compared to the critical current, decreasing L increases ic, while increasing iw decreases ic more rapidly for the smaller L values. As the temperature is raised, ic occurs at smaller and smaller magnetic flux in the loop. When the loop is positioned in such a way that no magnetic flux from the current in the wire is coupled to the loop, the current in the loop is still modified through the superconducting condensate in the branch.

Spin glasses with cubic anisotropy

The infinite‐range quantum spin glasses with cubic anisotropy (K) are studied using a combination of the imaginary‐time representation with the n‐replica approach and the thermofield dynamic method. Mean‐field theory phase diagrams in the temperature‐anisotropy plane (T,K) are presented for quantum spin numbers S ranging from 2 to 9/2. At T=0, the ground state is nonmagnetic and the spin‐glass order is absent for integer spin and large cubic anisotropy, the sign of which depends on S. For half‐integer S and sufficiently low temperatures the spin‐glass phase persists for arbitrary K.