Related papers: Scale Dependent Dimensionality

We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such scale is built-in within those transformations, and one does not need to…

The Planck mass and the cosmological constant determine the minimum and the maximum distances in the physical universe. A relativistic theory that takes into account a fundamental distance limit $\ell$ on par with the fundamental speed…

We consider further consequences of recently [1] revealed role of cosmological constant \Lambda as of a physical constant, along with the gravitational one to define the gravity i.e. the General Relativity and its low-energy limit. We now…

The fine-structure constant alpha approximately 1/137 is traditionally regarded as a fundamental dimensionless parameter. I argue instead that alpha is a scaled quantity that arises only where the structural scales contributed by classical…

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases…

We discuss an aspect of string theory which has been tackled from many different perspectives, but incompletely: the role of nonlocality in the theory and its relation to the geometric shape of the string. In particular, we will describe in…

Many 0/1 datasets have a very large number of variables; on the other hand, they are sparse and the dependency structure of the variables is simpler than the number of variables would suggest. Defining the effective dimensionality of such a…

The variational properties of the scalar so--called ``Universal'' equations are reviewed and generalised. In particular, we note that contrary to earlier claims, each member of the Euler hierarchy may have an explicit field dependence. The…

One of the deepest insights from the general theory of relativity is the relational nature of spacetime. While it is a generally agreed on that the nature of spacetime must be drastically different at the Planck scale, it has been a common…

City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature…

Scalar-tensor theories have taken on a key role in attempts to confront the growing open questions in standard cosmology. It is important to understand entirely their dynamics at perturbative level including any possible spatial dependence…

Complex structures can only form in a universe that allows for bound states. While this is clearly observed in three-dimensions, added degrees of freedom in a higher-dimensional space preclude the immediate assumption that binding…

The higher-dimensional generalization of Randall-Sundrum approach with additional positive curvature $n$-dimensional and Ricci-flat $m$-dimensional compuct subspaces is considered in pure gravity theory with metric of space-time and…

Device independent dimension witnesses provide a lower bound on the dimensionality of classical and quantum systems in a "black box" scenario where only correlations between preparations, measurements and outcomes are considered. We address…

The standard picture of viable higher-dimensional theories is that extra dimensions manifest themselves at short distances only, their effects being negligible at scales larger than some critical value. We show that this is not necessarily…

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical…

A possible way to resolve the singularities of general relativity is proposed based on the assumption that the description of space-time using commuting coordinates is not valid above a certain fundamental scale. Beyond that scale it is…

The consequences of considering the measure of integration in the action to be defined by degrees of freedom independent of the metric are studied. Models without the cosmological constant problem, new ways of spontaneously breaking scale…

We construct a class of scalar field models coupled to matter that lead to the dependence of masses and coupling constants on the ambient matter density. Such models predict a deviation of couplings measured on the Earth from values…