Related papers: A study on Quantization Dimension in complete metr…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…
We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy,…
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar $f$, dubbed as density function, which…
The intrinsic unsharpness of a quantum observable is studied by introducing the notion of resolution width. This quantification of accuracy is shown to be closely connected with the possibility of making approximately repeatable…
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…
The goal of this work is to study the space of continuous functions whose ergodic averages converge everywhere towards a continuous function. We will connect, as in the case of a metric study, the convergence of the ergodic averages and the…
Quantum technologies exploit entanglement to enhance various tasks beyond their classical limits including computation, communication and measurements. Quantum metrology aims to increase the precision of a measured quantity that is…
We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order $r$, including negative values of $r$. To this end, we use the concept of partition…
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the…
Endeavoring to formulate an exhaustive solution to the measurement problem in view of the theory of decoherence leads to a better understanding of the status of the collapse and of the emergence of classicality, thanks to a precise…
We argue that quantum theory in curved spacetime should be invariant under the continuous spacetime symmetries thaat are connected with the identity. For typical warped-product spacetimes, we prove that such invariance can be actually…
In order to test the canonical quantization programme for general relativity we introduce a reduced model for a real sector of complexified Ashtekar gravity which captures important properties of the full theory. While it does not…
For a partially ordered set $(S, \mathord\preceq)$, the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on $S$ that characterize the order $\preceq$. In this paper…
We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…
Unavoidable disturbance caused by a quantum measurement implies that the realizable subsequent measurements are getting limited after one performs some measurement. The obvious general limitation that one cannot circumvent by sequential or…
Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we…
We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of…
Many modern search domains comprise high-dimensional vectors of floating point numbers derived from neural networks, in the form of embeddings. Typical embeddings range in size from hundreds to thousands of dimensions, making the size of…
Let $\mu$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions…