Abstract:
This article blends the concepts of space-time from theoretical physics and Einstein’s Relativity Theory to discuss the spatio-temporal nature of distance education. By comparing and contrasting speed-of-light space travel with the speed of computer processing, the leap is made to consider the fourth dimension and its phenomena for the Web traveler. Learning events are compared with events in time to depict the theory presented.

Abstract:
We show that there is a unique extremal even unimodular lattice of dimension 48 which has an automorphism of order 5 of type 5-(8,16)-8. Since the three known extremal lattices do not admit such an automorphism, this provides a new example of an extremal even unimodular lattice in dimension 48.

Abstract:
We investigate the asymptotic behavior as $k \to +\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ of solutions of the equations $\Delta^2 u_k=V_k e^{4u_k}$ on $\Omega$, where $\Omega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. The corresponding 2-dimensional problem was studied by Br\'ezis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension 2.

Abstract:
This paper explains how a model of the universe can be constructed by incorporating time and space into geometry in a unique way to produce a 4-space dimension/1-time dimension model. The model can then show how dark matter can be the gravity that is produced by real matter that exists throughout our entire universe. The model can also show how dark energy is not an increase in energy that is causing the accelerated expansion of the universe, but is an accelerating decrease in matter throughout the universe as the stars and galaxies in the universe continue to convert matter into energy during their life cycles. And then the model can show how a fourth space dimension must exist in our universe to locate a point in space.

Abstract:
In different passages of his dialogues, Plato showed deep mathematically-based physical insights. Regrettably most readers overlooked the respective statements, or they utterly did not understand those hints since they were full of philological fallacious terms. Respectable translators misinterpreted such statements and therefore Plato's respective remarks were not recognized as substantial knowledge. Furthermore, Plato often supplemented such basic remarks by diffusely veiled and varied allusions that were often ironically hidden somewhere in his dialogues by inconspicuous double meanings. However, this mode of intentionally coded discrete communication was generally not understood because such irony is not to everyone's taste. However, the attempts to reconstruct Plato's system on the basis of admittedly individually interpreted double meanings lead to a conclusive mathematical-physical cyclical system of dimensions. Additionally it was possible to assign Plato's system of philosophical ideas analogously to this cyclical system. Plato took the verifiability of the mathematical-physical results as proof of the system of his ideas and finally as proof of his ethical creed, the unconditional trust in the 'all surmounting Good.'

Abstract:
For a class of one-dimensional linear elliptic fourth-order equations with homogeneous Dirichlet boundary conditions it is shown that a non-positive and non-vanishing right-hand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radially symmetric solutions in a ball or in an annulus. Several applications are given, including applications to nonlinear equations and eigenvalue problems.

Abstract:
Alice wants to send an arbitrary binary word to Bob. We show here that there is no problem for her to do that with only two bits. Of course, we consider here information like a signal in 4D.

Abstract:
We present two approaches, one homological and the other simplicial, for the investigation of dimension quotients of groups. The theory is illustrated, in particular, with a conceptual discussion of the fourth and fifth dimension quotients.