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pro vyhledávání: '"Balitskiy, Alexey"'
In this paper, we bring a new perspective to persistent homology by incorporating key concepts from metric geometry. For a given compact subset $X$ of a Banach space $Y$, we study the topological features appearing in family $N_\bullet(X\subset Y)$ o
Externí odkaz:
http://arxiv.org/abs/2403.13980
Autor:
Balitskiy, Alexey
The Urysohn d-width of a metric space quantifies how closely it can be approximated by a d-dimensional simplicial complex. Namely, the d-width of a space is at most w if it admits a continuous map to a d-complex with all fibers of diameter at most w.
No power law systolic freedom is possible for the product of mod $2$ systoles of dimension $1$ and codimension $1$. This means that any closed $n$-dimensional Riemannian manifold $M$ of bounded local geometry obeys the following systolic inequality:
Externí odkaz:
http://arxiv.org/abs/2206.01968
We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at mos
Externí odkaz:
http://arxiv.org/abs/2112.04594
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, wher
Externí odkaz:
http://arxiv.org/abs/2106.10429
We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a
Externí odkaz:
http://arxiv.org/abs/2009.04558
Publikováno v:
Journal f\"ur die reine und angewandte Mathematik (Crelles Journal), vol. 2021, no. 780, 2021, pp. 265-274
The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the $1$-w
Externí odkaz:
http://arxiv.org/abs/2008.07718
Autor:
Balitskiy, Alexey
If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two paral
Externí odkaz:
http://arxiv.org/abs/2003.06707
Autor:
Balitskiy, Alexey, Wellman, Julian
Publikováno v:
Sel. Math. New Ser. 26, 15 (2020)
Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $\text{Gr}^{\geq 0}(n,k)$. Any two plabic graphs for the same positroid cell c
Externí odkaz:
http://arxiv.org/abs/1902.01530
We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the "weak simplex conjecture". We discuss possible approaches to it, and state related
Externí odkaz:
http://arxiv.org/abs/1701.07986