Related papers: Quantitative combinatorial geometry for continuous…
We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carath\'eodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions.…
We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let $s$ be the Hausdorff dimension of the spectrum. For $V>20$, we show that the restriction of the…
The well known fractional Helly theorem and colorful Helly theorem can be merged into the so called colorful fractional Helly theorem. It states: For every $\alpha \in (0, 1]$ and every non-negative integer $d$, there is $\beta_{col} =…
We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also propose the…
The paper continues the author's research in the problem of quantitative investigation of basic curvelinear quasiinvariants of quasiconformal curves. It concerns polygons with infinite number of vertices and provides various distortion…
We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be…
Given $d+1$ sets of points, or colours, $S_1,\ldots,S_{d+1}$ in $\mathbb R^d$, a colourful simplex is a set $T\subseteq\bigcup_{i=1}^{d+1}S_i$ such that $|T\cap S_i|\leq 1$, for all $i\in\{1,\ldots,d+1\}$. The colourful Carath\'eodory…
The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a "dense" set $B_1$, for each $b_1\in B_1$ a "dense" set $B_2^{b_1}$ (depending on $b_1$), for each $b_2\in B_2^{b_1}$ a…
Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral…
It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi's diastasis and its domain of definition, the Euler-Poincar\'e characteristic, the number of…
We prove that the classical Menshov-Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces…
We prove in three ways the basic fact of analysis that complex derivatives are continuous. The first, classical, proof of Cauchy and Goursat uses integration. The second proof of Whyburn and Connell is topological and is based on the…
In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem.…
Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex…
The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger…
The article introduces Ahlfors' generalization of the Schwarz lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings,…
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
We develop a new cohomology theory for finite-dimensional left-symmetric color algebras and their finite-dimensional bimodules, establishing a connection between Lie color cohomology and left-symmetric color cohomology. We prove that the…
General covariant expressions for measurable angles, distances, velocities, and accelerations are provided in terms of fundamental parameters that can be applied in any setup. The relativistic aberration of light relationship is presented…
We study a particular Einstein-Maxwell-Dilaton black hole configuration with cosmological constant, expressed in terms of the curvature radius, from the point of view of quasi-homogeneous thermodynamics. In particular, we show that the…