Relativity and quantum theory are two epoch-making theories of 20th century physics, which not only established modern material civilization, but also changed people's world view. However, the Copenhagen interpretation of quantum mechanics remainsa controversial issue. The Copenhagen interpretation holds that the motion of the microscopic particles is uncertain, and the module square of the wave function is the probability density to discover a particle. As a matter of fact, the wave function is a concept much more abundant than probability density. A wave function or a spinor field completely describes all the properties of a fundamental particle and can define all the mechanical quantities of the particle, and the spinor equation logically implies both classicaland quantum mechanics of the particle. The wide application of abstract operator algebra in quantum theories further increases the mystery and puzzles, but the description of microscopic particles by operator algebra is incomplete. In this paper, we take spinor equation as the logical premise to establish and explain the basic principles and conclusions of the quantum theory, including the details of origin and limitations of the operator algebra, the logical relation between the spinor equation and Newtonian and quantum mechanics. We consistently derive all the correct principles and conclusions and suggest some new topics worthy of further study. These discussions should be important in clarifying the nature of the controversial issues.

The explanation of the perfect forms in nature is a subject which has attracted a great attention since the beginning of our civilization. E.g., despite the longstanding interest in the shapes of the eggs, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable parameters. Here we are exploring another geometrical model suggested by G. Brandt long time ago but which has not received any proper treatment. Actually, it is based on fourth order plane algebraic curve for which we have derived several parameterizations. The most interesting of them is the uniformization of the curve via the Jacobian elliptic functions. Some comparisons with the experimental results were made but this subject deserves further investigation.

Certain exact solutions (the so called functionally-invariant solutions) of the \(ABC\) equation and some two Martinez Alonso-Shabat equations, have been obtained by using the so-called structural decomposition method. Some of these solutions are localized.

The gradient-flow equations with respect to the potential functions in information geometry are reconsidered from the perspective of the Weyl integrable geometry. The pre-geodesic equations associated with the gradient-flow equations are regarded as the general pre-geodesic equations in the Weyl integrable geometry.