Related papers: A Theory of Dimension
Formal definitions of quantities, quantity spaces, dimensions and dimension groups are introduced. Based on these concepts, a theoretical framework and a practical algorithm for dimensional analysis are developed, and examples of…
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…
We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension…
The problem of advancing coordinatization of mathematics is considered. The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation…
A classical two dimensional theory of gravity which has a number of interesting features (including a Newtonian limit, black holes and gravitational collapse) is quantized using conformal field theoretic techniques. The critical dimension…
The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional…
We use experiment-supported dimensional analysis to further bolster our arguments that crucial information on the emergence and/or nature of space could be extracted from the combination of the properties of gravitational and strong…
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied.…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
We use double categories to obtain a single theorem characterizing certain exponentiable morphisms of small categories, topological spaces, locales, and posets.
We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…
We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of…
In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried using both, the topological algebra and its central extension, which arise from the twisting of $N=2$ supersymmetry in four…
Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in…
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…
Evidence for fine-tuning of physical parameters suitable for life can perhaps be explained by almost any combination of providence, coincidence or multiverse. A multiverse usually includes parts unobservable to us, but if the theory for it…
A theory of gravity in $d+1$ dimensions is dynamically generated from a theory in $d$ dimensions. As an application we show how $N$ dynamically coupled gravity theories can reduce the effective Planck mass.