Related papers: Scale Dependent Dimensionality
Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust…
Numerous approaches to quantum gravity report a reduction in the number of spacetime dimensions at the Planck scale. However, accepting the reality of dimensional reduction also means accepting its consequences, including a variable speed…
One often distinguishes between a line and a plane by saying that the former is one-dimensional while the latter is two. But, what does it mean for an object to have $d-$dimensions? Can we define a consistent notion of dimension rigorously…
This paper has been withdrawn by the author
Five fundamental scales of mass follow from holographic limitations, a self-similar law for angular momentum and the basic scaling laws for a fractal universe with dimension 2. The five scales correspond to the observable universe,…
A scale-dependent cosmology is proposed in which the Robertson-Walker metric and the Einstein equation are modified in such a way that $\Omega_0$, $H_0$ and the age of the Universe all become scale-dependent. Its implications on the…
In unified field theories with more than four dimensions, the form of the equations of physics in spacetime depends in general on the choice of coordinates in higher dimensions. The reason is that the group of coordinate transformations in…
We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation…
We compare limit-based and scale-local dimensions of complex distributions, particularly for a strange attractor of the Henon map. Scale-local dimensions as distributions on scale are seen to exhibit a wealth of detail. Limit-based…
A relativistic generalisation of a well-known method for approximating the dynamics of topological defects in condensed matter is constructed, and applied to the evolution of domain walls in a cosmological context. It is shown that there…
It has recently been proposed that the hierarchy problem can be solved by considering the warped fifth dimension compactified on $S^{1}/Z_{2}$. Many studies in the context have assumed a particular choice for an integration constant…
We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…
Over the past few years, evidence has begun to accumulate suggesting that spacetime may undergo a "spontaneous dimensional reduction" to two dimensions near the Planck scale. I review some of this evidence, and discuss the (still very…
We prove some basic results on the dimension theory of algebraic stacks, and on the multiplicities of their irreducible components, for which we do not know a reference.
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must…
Dimensional analysis is a simple qualitative method for determining essential connections between physical quantities. It is applicable to a multitude of physics problems, many of which canbe introduced early on in a university physics…
The relevance of the Planck scale to a theory of quantum gravity has become a worryingly little examined assumption that goes unchallenged in the majority of research in this area. However, in all scientific honesty, the significance of…
For decades, metrologists have debated heatedly whether a plane angle is a dimensional or dimensionless quantity; whether it is a base quantity in the International System of Units (SI) or a derived quantity. Two main points of view have…
It is time to renew old ways of thinking about dimensional analysis. Specifically, more than $n-r$ invariants and more than one functional relation between invariants need to be considered simultaneously. Thus generalized, dimensional…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…