Narrow Results

In this paper, we study the stationary and dynamical properties of three-dimensional trapped Bose–Einstein condensates with attractive interactions subjected to a random potential. To this end, a variational method is applied to solve the underlying Gross–Pitaevskii equation. We derive analytical predictions for the energy, the equilibrium width, and evolution laws of the condensate parameter. The breathing mode oscillations’ frequency of the condensate has been also calculated in terms of the gas and disorder parameters. We analyze in addition the dynamics of collapse from the Gaussian approximation. Surprisingly, we find that the intriguing interplay of the attractive interaction and disorder effects leads to prevent collapse of the condensate.

We investigate a question raised by Lempert–Henkin on the regularity of the equation on infinite dimensional spaces. More precisely, we improve slightly a classical example of Coeuré–Lempert.

The classical limit of wave quantum mechanics is analyzed. It is shown that the basic requirements of continuity and finiteness to the solution of the form ψ(x) = Ae^{i ϕ (x)} + Be^{-i ϕ (x)}, where and W(x) is the reduced classical action of the physical system, give the asymptote of the wave equation and general quantization condition for the action W(x), which yields the exact eigenvalues of the system.

In this paper, a novel parallel-coupled quadrature LC oscillator (PC-QO) is presented that can automatically eliminate imbalances of output phase and amplitude resulting from mismatches in the LC tanks. The method of using unequal coupling factors in a parallel-coupled quadrature oscillator is the base of this design; canceling the phase and amplitude errors takes place by tuning the imbalanced coupling factor using tuneable tail currents. First, the proposed circuit senses the amplitude error and then adjusts the coupling factors according to the situation. We show that using the inversely proportional coupling factors can eliminate the phase and amplitude error simultaneously. In other words, this design uses the amplitude control loop to cancel the phase and amplitude error. The circuit has been simulated using TSMC 0.18 CMOS practical model to confirm the high accuracy of the analysis and capability of the canceling $I$/$Q$ mismatch technique. The simulation results show that the phase noise of the proposed quadrature voltage controlled oscillator (QVCO) is −123.9 dBc/Hz at 1MHz offset from 4.4GHz operation frequency. The total power consumption of the QVCO is 3.4mW.

We show that the Diophantine problem (decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right-angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.

A macroscopic cable equation, which describes the passive linear ("electrotonic") response of a myelinated axon, was previously derived from a segmented cable equation using Keller's two-space homogenization method [Basser, PJ, Med. and Biol. Comput., 1993, Vol. 31, pp. S87–S92]. Here we use the space and length constants of this averaged cable equation to predict classical scaling laws that govern relationships among the inner and outer diameters of the axon's myelin sheath and the distance separating adjacent nodes of Ranvier. These laws are derived by maximizing the characteristic speed of an electrical disturbance along the axon, i.e., the ratio of the characteristic length and the characteristic time constants of the macroscopic cable, subject to the constraint that the nodal width is constant. Using this result, it is also possible to show that all myelinated axons are equally fault tolerant. No free parameters were used in this analysis; all variables and physical constants used in these calculations were taken from published experimental data.

We give a short proof to some results of Sun and Wang on counting the number of solutions of a class of equations over finite fields. Our proof is elementary and only based on matrix properties and recursion techniques.

Some real matrix representations of 4-dimensional Clifford numbers are presented and some properties of 4-dimensional Clifford numbers are obtained. As an application, an explicit expression of the general solution of equation axb = d and ax = xb are obtained in terms of a and b in 4-dimensional Clifford algebra Cℓ_0,3. As a by-product, we obtain the necessary and sufficient conditions for two Clifford numbers to be similar.