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### Cylinders' percolation: decoupling and applications [article]

Alves, Caio and Teixeira, Augusto
2021 arXiv   pre-print
Our work introduces a new notion of fast decoupling, proves that it holds for the model in question and finishes with an application.  ...  In this paper we establish a strong decoupling inequality for the cylinder's percolation process introduced by Tykesson and Windisch in arXiv:1010.5338 .  ...  Given a compact set A ⊂ Rd , we denote by LA the set of lines in L that intersect A.  ...

### Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem

Noga Alon, Daniel J Kleitman
It is shown that for every p > q > d + 1 there is a c = c(p, q, d) < cc such that for every family % of compact, convex sets in Rd which has the (p, q) property there is a set of at most c points in Rd  ...  A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection.  ...  ., for every family B of compact, convex sets in Rd which has the (p, q)-property there is a set of at most c points in Rd that intersects each member of F".  ...

### Subdivision methods for solving polynomial equations

B. Mourrain, J.P. Pavone
2009 Journal of symbolic computation
real roots in a box of R n .  ...  This paper presents a new algorithm for solving a system of polynomials, in a domain of R n .  ...  real roots in a box D ⊂ R n .  ...

### Gallai-type Results for Multiple Boxes and Forests

J. Lehel
1988 European journal of combinatorics (Print)
Foreveryintegerc,dandk;;;. min{c, d} + 1, thereexistsaconstantt(dependingonly on k, c and d) with the property: if Fis any family of d-dimensional c-boxes such that every k members have a common point  ...  In case of families composed by the unions of c d-dimensional boxes, we prove that the minimum value oft is finite for fixed c, dand k, iff k ;;. min {c, d} + l.  ...  Rd is called ad-dimensional c-box if it is the union of c (or less) d-dimensional boxes.  ...

### Page 6476 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews
property.  ...  Summary: “In this paper, the refined box dimension of a class of surfaces generated by a boundary value problem of a fourth-order difference equation is studied.  ...

### Characterization of the graphs with boxicity ⩽2

Martin Quest, Gerd Wegner
1990 Discrete Mathematics
Following Roberts [4] the boxicity b(G) of a graph G is defined as the smallest d such that G is the intersection graph of boxes in Euclidean d-space, i.e. parallelepipeds with edges parallel to the coordinate  ...  The intersection graph of a family 8TI of sets has the sets in %' as vertices and an edge between two sets iff they have nonempty intersection.  ...  First note that 58 has (as any family of boxes in Rd) the Helly-l-property: If B' E '\$j and Bi rl Bk # 0 for all pairs Bi, B, E B', then nBEmP B # 0.  ...

### Geometric medians

Joseph Gil, William Steiger, Avi Wigderson
1992 Discrete Mathematics
We discuss several generalizations of the notion of median to points in Rd. They arise in Computational Geometry and in Statistics.  ...  These notions are compared with respect to some of their mathematical properties. We also consider computational aspects.  ...  A distinctive property of the box median is that it always has quadratic depth. Lemma 1. There is a positive constant c(d) c i such that for every set S c Rd with n points, /3* > c(d)n'. Proof.  ...

### The Hausdorff measure functions: A new way to characterize fractal sets

Jun Li, Fahima Nekka
2003 Pattern Recognition Letters
The HMSF does carry a huge amount of information about the set to likely be explored in a chosen way. Depending on the nature of the set, we propose two ways to extract this information.  ...  We apply these methods to typical fractals as well as to synthetic models of porous media. This results in a complete distinction between same fractal dimension sets.  ...  Acknowledgements We are very grateful to Professor Jacques Levy-Vehel for his helpful hints and assistance for some aspects of this project. We also thank the unknown referee for his comments.  ...

### Covering with Euclidean Boxes

Imre Bárány, Jenö Lehel
1987 European journal of combinatorics (Print)
Parallelopipeds of the d-dimensional Euclidean space ~d with faces parallel to the axes are called boxes and box (p, q) denotes the intersection of all boxes containing the points p, q E ~d.  ...  Here we prove the existence of a constant c depending only on d such that any compact set V c ~d has a subset of cardinality at most c which induces a box-cover of V, that is V c Up.qes box(p, q).  ...  It is easy to see and is actually well-known [10] that in a cone-subdivision every cone is a polyhedral set, that is the intersection of a finite family of closed half-spaces of !Rd.  ...

### Subadult Virtual Anthropology Database (SVAD) Data Collection Protocol: Cranial Landmarks and Craniometrics

M Kate Spradley, Christopher A Wolfe, Kyra E Stull, Elaine Y Chu, Kristen A Broehl, Tatiana Vlemincq-Mendieta, Marin A Pilloud, G Richard Scott, Louise K Corron
2021 Zenodo
This protocol was developed as part of National Institute of Justice Award 2019 DU-BX-0039. The new version (v2) has been updated to include corrected figures.  ...  This document presents the standardized protocol for placing landmarks on virtual 3D renderings of subadult crania using the Amira visualization software.  ...  In the "Properties" pane, click on the [Edit box] of the "Colormap" line. You can scroll down the menu and choose one of the Colormap options. You can revert back if necessary at any point.  ...

### Scaling property and peculiar velocity of global monopoles

Masahide Yamaguchi
2002 Physical Review D, Particles and fields
We investigate the scaling property of global monopoles in the expanding universe.  ...  By directly solving the equations of motion for scalar fields, we follow the time development of the number density of global monopoles in the radiation dominated (RD) universe and the matter dominated  ...  This work was partially supported by the Japanese Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology.  ...

### Irrepresentability by multiple intersection, or why the interval number is unbounded

Edward R. Scheinerman
1985 Discrete Mathematics
We are especially interested in the case where the given family of sets generalizes the family of real intervals. We extend our results to uniform hypergraphs and simplicial complexes.  ...  We consider the following question: Given a family of sets, is there a positive integer, t, so that every graph is the intersection graph of sets each of which is the union of t sets from the given family  ...  Acknowledgments The material presented in this paper has also appeared in the author's docotoral dissertation [8] written at Princeton University under the supervision of Douglas West.  ...

### Principal specializations of Schubert polynomials and pattern containment [article]

Yibo Gao
2019 arXiv   pre-print
We show that the principal specialization of the Schubert polynomial at w is bounded below by 1+p_132(w)+p_1432(w) where p_u(w) is the number of occurrences of the pattern u in w, strengthening a previous  ...  We then make a conjecture relating the principal specialization of Schubert polynomials to pattern containment.  ...  Dually, if (i, j), (i + 1, j) ∈ D, then their corresponding boxes in RD(w) lie in the same column and that there are no other boxes in between. Proof of Theorem 4.1.  ...

### Formalizing natural‐language spatial relations between linear objects with topological and metric properties

Jun Xu
2007 International Journal of Geographical Information Science
models of spatial relations in natural language.  ...  Two groups of metric details were derived: splitting ratios, which are the normalized values of lengths and areas of intersections; and closeness measures, which are the normalized distances between disjoint  ...  The wrong query of 'parallel to'. properties between two linear objects, and the values of the 9-intersection model were used to describe the topological properties between them.  ...

### On Combinatorics and Topology of Pairwise Intersections of Schubert Cells in SL n /B [chapter]

Boris Shapiro, Michael Shapiro, Alek Vainshtein
1997 The Arnold-Gelfand Mathematical Seminars
It is shown in [Sh] that the intersection RC 1,w0 ∩ RC w0,w0 enjoys such a property. Conjecture 6. Any intersection RC 1,α ∩ RC σ,β enjoys the M-property.  ...  Topological properties of intersections of pairs and, more generally, of k-tuples of Schubert cells belonging to distinct Schubert cell decompositions of a flag space are of particular importance in representation  ...  that is the minimal element in the lexicographic linear order is one of the strata of the minimal dimension. Conjecture 3.  ...
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