Radon Number Research Articles

Bounding Radon Numbers via Betti Numbers

Abstract We prove general topological Radon-type theorems for sets in $\mathbb R^{d}$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon $-nets as well as a $(p,q)$-theorem for those sets. More precisely, given a family ${\mathcal{F}}$ of subsets of ${\mathbb{R}}^{d}$, we will measure the homological complexity of ${\mathcal{F}}$ by the supremum of the first $\lceil d/2\rceil $ reduced Betti numbers of $\bigcap{\mathcal{G}}$ over all nonempty ${\mathcal{G}} \subseteq{\mathcal{F}}$. We show that if ${\mathcal{F}}$ has homological complexity at most $b$, the Radon number of ${\mathcal{F}}$ is bounded in terms of $b$ and $d$. In case that ${\mathcal{F}}$ lives on a surface and the number of connected components of $\bigcap \mathcal G$ is at most $b$ for any $\mathcal G\subseteq \mathcal F$, then the Radon number of ${\mathcal{F}}$ is bounded by a function depending only on $b$ and the surface itself. For surfaces, if we moreover assume the sets in ${\mathcal{F}}$ are open, we show that the fractional Helly number of $\mathcal F$ is linear in $b$. The improvement is based on a recent result of the author and Kalai. Specifically, for $b=1$ we get that the fractional Helly number is at most three, which is optimal. This case further leads to solving a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.

Tolerant Radon partitions on the all-paths convexity in graphs

In a graph G, the all-paths transit function A(u, v), consists of the set of all vertices in the graph G which lies in some path connecting u and v. Convexity obtained by the all-paths transit function is called all-paths convexity. A partition of a set P of vertices of a graph G into two disjoint non-empty subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition (P0, Q0) of P into nonempty subsets is called t-tolerant Radon partition, if for any set S ⊆ P with | S | ≤ t , the intersection of the convex hulls 〈 P 0 − S 〉 ∩ 〈 Q 0 − S 〉 ≠ ∅ . Here we study the t-tolerant Radon number of G with respect to all-paths convexity. It is the minimum number k such that any collection of k vertices of G has t-tolerant Radon partition. We prove that if G has only one block, then k = 2 t + 2 whenever t ≥ 1 . Finally, we prove that if the block-cut vertex tree representation of G is a path then k = 2 t + 3 , and if the block-cut vertex tree representation of G is a tree that is not a path then k = 2 t + 4 .

Convex invariants of M-fuzzifying convex spaces

The numbers of Helly, Carathéodory and Radon are a central theme in classic convex spaces and each of them is invariant under isomorphism. The theory of M-fuzzifying convex spaces is an active research field since it is presented. This paper focus on convex invariants of M-fuzzifying convex spaces. The degrees of Helly independent, Carathéodory independent and Radon independent of non-empty set are defined in the framework of M-fuzzifying convex spaces. By those definitions, we introduce the Helly number, Carathéodory number and Radon number of M-fuzzifying convex spaces. Finally, we inspect M-fuzzifying topology and M-fuzzifying JHC which are characterized by convex invariants of M-fuzzifying convex spaces.

Radon Numbers Grow Linearly

Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_kle c(r_2)cdot k.

Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

Radon numbers and the fractional Helly theorem

A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we answer a question of Kalai, by showing a fractional Helly theorem for convexity spaces with bounded Radon number. As a consequence we also get a weak ε-net theorem for convexity spaces with bounded Radon number. This answers a question of Bukh and extends a recent result of Moran and Yehudayoff.

On Weak $$\epsilon $$-Nets and the Radon Number

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak \(\epsilon \)-nets.

The convexity of induced paths of order three and applications: Complexity aspects

In this paper, we introduce a new convexity on graphs similar to the well known P3-convexity, which we will call P3∗-convexity. We show that several P3∗-convexity parameters (hull number, convexity number, Carathéodory number, Radon number, interval number and percolation time) are NP-hard even on bipartite graphs. We prove a strong relationship between this convexity and the well known geodesic convexity, which implies several NP-hardness results for the latter. In order to show that, we prove that the hull number for the P3-convexity is NP-hard even for subgraphs of grids and that the convexity number for the P3-convexity is NP-hard even for bipartite graphs with diameter 3. We also obtain linear time algorithms to determine those parameters for the above mentioned convexities for cographs and P4-sparse graphs.

DETERMINATION OF RADON CONCENTRATIONS IN DWELLING IN ACEH

Abstract. Determination of radon concentrations in dwelling in Aceh region by using a passive method has been conducted. In this research, area considered was divided into several sections called grid. Each grid represents an area of 60 km x 60 km in which, depend on public response, 6-10 passive radon monitors were installed. The number of passive radon monitors installed in Aceh is 200 units, and they can be taken back as many as 191 units or 95.50 %. The passive radon monitors have stayed in dwelling for 3-4 months and after period of the exposure, those radon monitors were taken back and brought to laboratory for further process, and then the track were read and the radon concentrations were calculated. Furthermore, data of radon concentration in dwelling and GPS location were put into MapInfo Software v.10.5 to create a map of radon concentration. The results of the analysis of the radon concentration in dwelling in Aceh demonstrate that the concentrations are in the range of 3.32 ± 0.23 Bq/m3 up to 68.30 ± 4.83 Bq/m3. This result was lower than the radon reference level determined by UNSCEAR, which was 300 Bq/m3. The data are useful in the regional extension and development plans, as well as the basis for health policy analysis due to the existence of radon in Indonesia. Furthermore, these data will become the contribution of Indonesia in the international world through UNSCEAR, IAEA and WHO. The data obtained can be used as partial data in creating a map of radon concentration in residents’ houses in Aceh, as a part of the map of radon concentration in Indonesia. Keywords: radon concentration, dwelling, Aceh, passive methodREFERENSI UNSCEAR, 1996, Natural Radiation Exposures, Forty Fifth Session, VienaIAEA, 2005, Radiation, People and the Environment, Viena.Bunawas, Emlinarti, M. Affandi, 1996, Penentuan laju lepasan radon dari bahan bangunan menggunakan metode pasip dengan metode jejak nuklir, Prosiding PPIKRL, PSPKR-BATAN, 20-21 Agustus 1996, pp. 16-21.Sutarman, L. Nirwani, Emlinarti dan A. Warsona, 2005, Penentuan konsentrasi gas radon dan thoron menggunakan detektor film LR-115 di DKI Jakarta dan sekitarnya, Prosiding PPI–PDIPTN P3TM-BATAN, Jogjakarta, p. 212-221.M. Affandi, D. Iskandar, dan Bunawas, 1996, Radon di Kompleks Perumahan BATAN, Presiding PIKRL, PSPKR-BATAN, p. 262-265Wahyudi, Kusdiana and D. Iskandar, 2016, Mapping of Indoor Radon Concentration in Houses Located in South Sulawesi Province, 2nd International Conference on the SERIR2 14th Biennial Conference of the SPERA, Bali, CTRSM-BATAN, p. 35-38.E. Pudjadi, Wahyudi, A. Warsona and Syarbaini, 2016, Measurement of Indoor Radon-Thoron Concentration in Dwellings of Bali Island, Indonesia, 2nd International Conference on the SERIR2 14th Biennial Conference of the SPERA, Bali, CTRSM-BATAN, p. 186-192.M.H.Magalhães, et al., 2003. Radon-222 in Brazil: an outline of indoor and outdoor measurements. Journal of Environmental Radioactivity, 67(2), pp.131–143.F.S. Al-Saleh, 2007. Measurements of indoor gamma radiation and radon concentrations in dwellings of Riyadh city, Saudi Arabia. Applied Radiation and Isotopes, 65(7), pp.843–848.

Inapproximability results and bounds for the Helly and Radon numbers of a graph

Let C be a convexity on a set X and denote the convex hull of S⊆X in C by H(S). The Helly number (Radon number) of C is the minimum integer k such that, for every S⊆X with at least k+1 elements, it holds ⋂v∈SH(S∖{v})≠∅ (there is a bipartition of S into sets S1 and S2 with H(S1)∩H(S2)≠∅). In this work, we show that there is no approximation algorithm for the Helly or the Radon number of a graph G of order n in the geodetic convexity to within a factor n1−ε for any ε>0, unless P = NP, even if G is bipartite. Furthermore, we present upper bounds for both parameters in the geodetic convexity of bipartite graphs and characterize the families of graphs achieving the bound for the Helly number.

Carathéodory, Helly, and Radon Numbers for Sublattice and Related Convexities

The Carathéodory, Helly, and Radon numbers are three main invariants in convexity theory. These invariants have been determined, exactly or approximately, for a number of different convexity structures. We consider convexity structures defined by the sublattices and by the convex sublattices of finite-dimensional Euclidian, integer, and Boolean spaces. Such sublattices arise in submodular optimization (lattice programming) and in monotone comparative statics of optimization and fixed point problems. We also consider integral L-natural convexities, induced by dual network flow constraint systems. We determine the exact Carathéodory, Helly, and Radon numbers of most of these convexities, and very close upper and lower bounds for the other Carathéodory numbers. Our results imply, for example, that if a set can be obtained with unions and intersections from a given family of subsets of a finite set then it can be obtained with unions and intersections from a small subfamily. We also show that finding the Carathéodory number of integral L-natural convexities reduces to an extremal problem in the theory of permutations, solved in a companion paper. We leave as open problems the determination of the Helly and Radon numbers of the integer convex sublattice convexity.

Geodetic convexity parameters for [formula omitted]-graphs

Following a suggestion of Campos et al. (2015) we show that, within the geodetic convexity, the interval number, the convexity number, the Carathéodory number, and the Radon number can be computed in polynomial time for (q,q−4)-graphs.

Radon numbers for trees

We consider P3-convexity on graphs, where a set U of vertices in a graph G is convex if every vertex not in U has at most one neighbour in U.Tverberg’s theorem states that every set of (k−1)(d+1)+1 points in Rd can be partitioned into k sets with intersecting convex hulls. As a special case of Eckhoff’s conjecture, we show that a similar result holds for P3-convexity in trees.A set U of vertices in a graph G is free if no vertex of G has more than one neighbour in U. We prove an inequality relating the Radon number for P3-convexity in trees with the size of a maximum free set.

Inapproximability results for graph convexity parameters

In this paper, we prove several inapproximability results on the P3-convexity and the geodesic convexity in graphs. We prove that determining the P3-hull number and the geodetic hull number are APX-hard problems. We prove that the Carathéodory number, the Radon number and the convexity number of both convexities are O(n1−ε)-inapproximable in polynomial time for every ε>0, unless P=NP. We also prove that the interval numbers of both convexities are W[2]-hard and O(log⁡n)-inapproximable in polynomial time, unless P=NP. Moreover, these results hold for bipartite graphs in the P3-convexity.

Near-linear-time algorithm for the geodetic Radon number of grids

The Radon number of a graph is the minimum integer r such that all sets of at least r of its vertices can be partitioned into two subsets whose convex hulls intersect. Determining the Radon number of general graphs in the geodetic convexity is NP-hard. In this paper, we show the problem is polynomial for d-dimensional grids, for all d≥1. The proposed algorithm runs in near-linear O(d(logd)1/2) time for grids of arbitrary sizes, and in sub-linear O(logd) time when all grid dimensions have the same size.

Some Steiner concepts on lexicographic products of graphs

Let G be a graph and W a subset of V(G). A subtree with the minimum number of edges that contains all vertices of W is a Steiner tree for W. The number of edges of such a tree is the Steiner distance of W and union of all vertices belonging to Steiner trees for W form a Steiner interval. We describe both of these for the lexicographic product of graphs. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number.

Complexity aspects of the computation of the rank of a graph

Special issue PRIMA 2013 We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.

The convexity of induced paths of order three

In this paper, we introduce a new convexity on graphs similar to the well known P3-convexity [R.Barbosa, E.Coelho, M.Dourado, D.Rautenbach, J.L.Szwarcfiter, A.Toman. On the Radon number for P3-convexity, Proc. LATINʼ2012, Lecture Notes in Computer Science7256 (2012), 267–278], which we will call P3⁎-convexity. We show that several P3⁎-convexity parameters (hull number, convexity number, Carathéodory number, Radon number, interval number and percolation time) are NP-hard even on bipartite graphs. We show a strong relation between this convexity and the well known geodesic convexity [M.Dourado, F.Protti, D.Rautenbach and J.L.Szwarcfiter. Some remarks on the geodetic number of a graph, Discrete Mathematics310 (2010), 832–837], which implies several NP-hardness results on the geodesic convexity. We also obtain linear time algorithms to determine all those parameters on the above mentioned convexities for some graph classes.

Polynomial time algorithm for the Radon number of grids in the geodetic convexity

The Radon number of a graph is the minimum integer r such that all sets of at least r vertices of the graph can be partitioned into two subsets whose convex hulls intersect. We present a near-linear O(dlogd) time algorithm to calculate the Radon number of d-dimensional grids in the geodetic convexity. To date, no polynomial time algorithm was known for this problem.

Algorithmic and structural aspects of the P 3-Radon number

The generalization of classical results about convex sets in ℝ n to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P 3-convexity on graphs. P 3-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon’s classical convexity result. We establish hardness results and describe efficient algorithms for trees.