Related papers: Quantization dimensions of compactly supported pro…
We study several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. We prove new bounds on these dimensions and raise several questions for further investigation. For instance, we…
An upper bound for the Kantorovich transport distance between probability measures on multidimensional Euclidean spaces is given in terms of transport distances between one dimensional projections. This quantifies the Cram\'er-Wold…
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct…
Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi…
We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For…
In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal U}) \le…
A characterization is presented of barycenters of the Radon probability measures supported on a closed convex subset of a given space. A case of particular interest is studied, where the underlying space is itself the space of finite signed…
Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial…
One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost…
Analyzing the dimension of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. In this paper, we consider this…
Recently, a new notion of quantum R\'enyi divergences has been introduced by M\"uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J.Math.Phys. 54:122203, (2013), and Wilde, Winter, Yang, Commun.Math.Phys. 331:593--622, (2014), that has…
We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair…
The 4-dimensional space-time is extended to pseudo-complex coordinates. Proposing the standard quantization rules in this extended space, the ones for the 4-dimensional sub-space acquire, as one solution, the commutation relations with…
A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…
The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein…
In this paper we show that if $\mu$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_\mu$ is bounded in $L^2(\mu)$, and $B\subset\mathbb R^{n+1}$ is a ball with $\mu(B)\approx…
We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.
Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this paper, we…
A novel measure, quantumness of correlations is introduced here for bipartite states, by incorporating the required measurement scheme crucial in defining any such quantity. Quantumness coincides with the previously proposed measures in…
A new global approach in the study of duality transformations is introduced. The geometrical structure of complex line bundles is generalized to higher order U(1) bundles which are classified by quantized charges and duality maps are…