Equivalence of String Classical and Quantum Energy beside Equivalence of Wave Packet and Relativistic Velocity in Eucleadian and Curved Space

Plank quantum and classical string energy relations seem to be uncorrelated. This work correlated them. The relativistic energy-momentum relation has been used together with plank and de Brogglie hypothesis to prove that the wave group velocity is equal to the particle velocity in both ordinary and curved space. The plank energy relation is shown also to be related to the classical energy relation of an oscillating string. Starting from plank energy relation for n photons and performing integration, the expression of classical string energy was obtained. This means that one can treat electromagnetic waves as a collection of continuous photons having frequencies ranging from zero to w. Conversely, starting from classical string energy relation by differentiating it with respect to angular frequency, the plank quantum energy for n photons has been found. This means that the quanta results from separation of electromagnetic waves to single isolated waves. Each wave consists of n photons or quanta.

Dynamical Study of a Constant Viscous Dark Energy Model in Classical and Loop Quantum Cosmology

Dynamical behaviors and stability properties of a flat space Friedmann-Robertson-Walker universe filled with pressureless dark matter and viscous dark energy are studied in the context of standard classical and loop quantum cosmology. Assuming that the dark energy has a constant bulk viscosity, it is found that the bulk viscosity effects influence only the quintessence model case leading to the existence of a viscous late time attractor solution of de- Sitter type, whereas the quantum geometry effects influence the phantom model case where the big rip singularity is removed. Moreover, our results of the Hubble parameter as a function of the redshift are in good agreement with the more recent data.

Conservation of Energy in Classical Mechanics and Its Lack from the Point of View of Quantum Theory

It is pointed out that the property of a constant energy characteristic for the circular motions of macroscopic bodies in classical mechanics does not hold when the quantum conditions for the motion are applied. This is so because any macroscopic body—lo-cated in a high-energy quantum state—is in practice forced to change this state to a state having a lower energy. The rate of the energy decrease is usually extremely small which makes its effect uneasy to detect in course of the observations, or experiments. The energy of the harmonic oscillator is thoroughly examined as an example. Here our point is that not only the energy, but also the oscillator amplitude which depends on energy, are changing with time. In result, no constant positions of the turning points of the oscillator can be specified;consequently the well-known variational procedure concerning the calculation of the action function and its properties cannot be applied.

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

Benard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating 1 solution with expansions up to any order are given and the convergence rates O（εm＋1/2）and the optimal convergence rates O（εm＋1） are obtained respectively. This improves the result of J.G. SHI.