Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature. © 2014 Elsevier Inc

(n,m)-fold covers of spheres

A well-known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere Sd is covered with less than d + 2 open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the d-dimensional sphere n times, with the additional property that the northern hemisphere is covered m > n times. We prove that if the open northern hemisphere is to be covered m times, then at least ⌈(d − 1)/2⌉ + n + m and at most d + n + m sets are needed. For the case of n = 1 and d ≥ 2, this number is equal to d + 2 if m ≤ ⌊d/2⌋ + 1 and equal to ⌊(d − 1)/2⌋ + 2 + m if m > ⌊d/2⌋ + 1. If the closed northern hemisphere is to be covered m times, then d + 2m − 1 sets are needed; this number is also sufficient. We also present results on a related problem of independent interest. We prove that if Sd is covered n times with open sets not containing a pair of antipodal points, then there exists a point that is covered at least ⌈d/2⌉ + n times. Furthermore, we show that there are covers in which no point is covered more than n + d times. © 2015, Pleiades Publishing, Ltd

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem

Tverberg-type theorems for intersecting by rays

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set

Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

Analogues of the central point theorem for families with dd-intersection property in Rd\mathbb R^d

In this paper we consider families of compact convex sets in $\mathbb R^d$ such that any subfamily of size at most $d$ has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families

Характеристика сили нервових процесів у плавців

OBJECTIVE: Higher levels of the novel inflammatory marker pentraxin 3 (PTX3) predict cardiovascular mortality in patients with chronic kidney disease (CKD). Yet, whether PTX3 predicts worsening of kidney function has been less well studied. We therefore investigated the associations between PTX3 levels, kidney disease measures and CKD incidence. METHODS: Cross-sectional associations between serum PTX3 levels, urinary albumin/creatinine ratio (ACR) and cystatin C-estimated glomerular filtration rate (GFR) were assessed in two independent community-based cohorts of elderly subjects: the Prospective Investigation of the Vasculature in Uppsala Seniors (PIVUS, n = 768, 51% women, mean age 75 years) and the Uppsala Longitudinal Study of Adult Men (ULSAM, n = 651, mean age 77 years). The longitudinal association between PTX3 level at baseline and incident CKD (GFR &lt;60 mL( ) min(-1)  1.73 m(-) ²) was also analysed (number of events/number at risk: PIVUS 229/746, ULSAM 206/315). RESULTS: PTX3 levels were inversely associated with GFR [PIVUS: B-coefficient per 1 SD increase -0.16, 95% confidence interval (CI) -0.23 to -0.10, P &lt; 0.001; ULSAM: B-coefficient per 1 SD increase -0.09, 95% CI -0.16 to -0.01, P &lt; 0.05], but not ACR, after adjusting for age, gender, C-reactive protein and prevalent cardiovascular disease in cross-sectional analyses. In longitudinal analyses, PTX3 levels predicted incident CKD after 5 years in both cohorts [PIVUS: multivariable odds ratio (OR) 1.21, 95% CI 1.01-1.45, P &lt; 0.05; ULSAM: multivariable OR 1.37, 95% CI 1.07-1.77, P &lt; 0.05]. CONCLUSIONS: Higher PTX3 levels are associated with lower GFR and independently predict incident CKD in elderly men and women. Our data confirm and extend previous evidence suggesting that inflammatory processes are activated in the early stages of CKD and drive impairment of kidney function. Circulating PTX3 appears to be a promising biomarker of kidney disease

Non-Convex Representations of Graphs

We show that every plane graph admits a planar straight-line drawing in which all faces with more than three vertices are non-convex polygon

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans