Recent work at La Laguna in Central Mexico provides an excellent illustration of the way in which... more Recent work at La Laguna in Central Mexico provides an excellent illustration of the way in which information from architecture, food remains, ceramic vessels and chemical signatures can be brought together to demonstrate communal feasting associated with specific structures and public spaces. Structure 12M-3 contained a range of evidence indicative of food preparation and consumption. Ritual effigy vessels depicted deities connected with food and fertility, and fire and the hearth. Taken together, the several lines of evidence indicate that Structure 12M-3 was a special building, located directly behind the main temple and devoted to the preparation and production of communal feasts that were held in the adjacent plaza. This provides new insights into community life in the urban centres of early Mesoamerica.
ABSTRACT Given a number \(P\) , we study the following three isoperimetric problems introduced by... more ABSTRACT Given a number \(P\) , we study the following three isoperimetric problems introduced by Besicovitch in 1952: (1) Let \(S\) be a set of \(n\) points in the plane. Among all the curves with perimeter \(P\) that enclose \(S\) , what is the curve that encloses the maximum area? (2) Let \(Q\) be a convex polygon with \(n\) vertices. Among all the curves with perimeter \(P\) contained in \(Q\) , what is the curve that encloses the maximum area? (3) Let \(r_{\circ }\) be a positive number. Among all the curves with perimeter \(P\) and circumradius \(r_{\circ }\) , what is the curve that encloses the maximum area? In this paper, we provide a complete characterization for the solutions to Problems 1, 2 and 3. We show that there are cases where the solution to Problem 1 cannot be computed exactly. However, it is possible to compute in \(O(n \log n)\) time the exact combinatorial structure of the solution. In addition, we show how to compute an approximation of this solution with arbitrary precision. For Problem 2, we provide an \(O(n\log n)\) -time algorithm to compute its solution exactly. In the case of Problem 3, we show that the problem can be solved in constant time. As a side note, we show that if \(S\) is a set of \(n\) points in the plane, then finding the area of the curve of perimeter \(P\) that encloses \(S\) and has minimum area is NP-hard.
Annual Symposium on Computational Geometry - SOCG'14, 2014
ABSTRACT Given a polygonal region containing a target point (which we assume is the origin), it i... more ABSTRACT Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.
ABSTRACT Let $\phi$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-e... more ABSTRACT Let $\phi$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-empty subset $\phi(A)$ of $\mathbb{R}^2$. A $\phi$-cover of a set $T=\{T_1, T_2, \dots, T_m\}$ of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree $T_i$ is contained in some region of the cover, and each region of the cover is either (1) $\phi(T_i)$ for some $i$, or (2) $\phi(A \cup B)$, where $A$ and $B$ are constructed by either (1) or (2), and $A \cap B \neq \emptyset$. We present two properties for the function $\phi$ that make the $\phi$-cover well-defined. Examples for such functions $\phi$ are the convex hull and the axis-aligned bounding box. For both of these functions $\phi$, we show that the $\phi$-cover can be computed in $O(n\log^2n)$ time, where $n$ is the total number of vertices of the trees in $T$.
ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minim... more ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n+m)logω) time, where ω=min{n,m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m+nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n+m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(nlogm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.
ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minim... more ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n+m)logω) time, where ω=min{n,m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m+nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n+m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(nlogm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.
ABSTRACT In this paper, we show that the θ-graph with three cones is connected. We also provide a... more ABSTRACT In this paper, we show that the θ-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.
A facile and high-yielding synthesis for a new family of sterically hindered heteroscorpionate li... more A facile and high-yielding synthesis for a new family of sterically hindered heteroscorpionate lithium salts based on an amidinate fragment [Li (NNN)(THF)] is described. Subsequent hydrolysis produces the corresponding amidine ligands. Furthermore, reaction of the ...
Recent work at La Laguna in Central Mexico provides an excellent illustration of the way in which... more Recent work at La Laguna in Central Mexico provides an excellent illustration of the way in which information from architecture, food remains, ceramic vessels and chemical signatures can be brought together to demonstrate communal feasting associated with specific structures and public spaces. Structure 12M-3 contained a range of evidence indicative of food preparation and consumption. Ritual effigy vessels depicted deities connected with food and fertility, and fire and the hearth. Taken together, the several lines of evidence indicate that Structure 12M-3 was a special building, located directly behind the main temple and devoted to the preparation and production of communal feasts that were held in the adjacent plaza. This provides new insights into community life in the urban centres of early Mesoamerica.
ABSTRACT Given a number \(P\) , we study the following three isoperimetric problems introduced by... more ABSTRACT Given a number \(P\) , we study the following three isoperimetric problems introduced by Besicovitch in 1952: (1) Let \(S\) be a set of \(n\) points in the plane. Among all the curves with perimeter \(P\) that enclose \(S\) , what is the curve that encloses the maximum area? (2) Let \(Q\) be a convex polygon with \(n\) vertices. Among all the curves with perimeter \(P\) contained in \(Q\) , what is the curve that encloses the maximum area? (3) Let \(r_{\circ }\) be a positive number. Among all the curves with perimeter \(P\) and circumradius \(r_{\circ }\) , what is the curve that encloses the maximum area? In this paper, we provide a complete characterization for the solutions to Problems 1, 2 and 3. We show that there are cases where the solution to Problem 1 cannot be computed exactly. However, it is possible to compute in \(O(n \log n)\) time the exact combinatorial structure of the solution. In addition, we show how to compute an approximation of this solution with arbitrary precision. For Problem 2, we provide an \(O(n\log n)\) -time algorithm to compute its solution exactly. In the case of Problem 3, we show that the problem can be solved in constant time. As a side note, we show that if \(S\) is a set of \(n\) points in the plane, then finding the area of the curve of perimeter \(P\) that encloses \(S\) and has minimum area is NP-hard.
Annual Symposium on Computational Geometry - SOCG'14, 2014
ABSTRACT Given a polygonal region containing a target point (which we assume is the origin), it i... more ABSTRACT Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.
ABSTRACT Let $\phi$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-e... more ABSTRACT Let $\phi$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-empty subset $\phi(A)$ of $\mathbb{R}^2$. A $\phi$-cover of a set $T=\{T_1, T_2, \dots, T_m\}$ of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree $T_i$ is contained in some region of the cover, and each region of the cover is either (1) $\phi(T_i)$ for some $i$, or (2) $\phi(A \cup B)$, where $A$ and $B$ are constructed by either (1) or (2), and $A \cap B \neq \emptyset$. We present two properties for the function $\phi$ that make the $\phi$-cover well-defined. Examples for such functions $\phi$ are the convex hull and the axis-aligned bounding box. For both of these functions $\phi$, we show that the $\phi$-cover can be computed in $O(n\log^2n)$ time, where $n$ is the total number of vertices of the trees in $T$.
ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minim... more ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n+m)logω) time, where ω=min{n,m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m+nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n+m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(nlogm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.
ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minim... more ABSTRACT We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n+m)logω) time, where ω=min{n,m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m+nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n+m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(nlogm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.
ABSTRACT In this paper, we show that the θ-graph with three cones is connected. We also provide a... more ABSTRACT In this paper, we show that the θ-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.
A facile and high-yielding synthesis for a new family of sterically hindered heteroscorpionate li... more A facile and high-yielding synthesis for a new family of sterically hindered heteroscorpionate lithium salts based on an amidinate fragment [Li (NNN)(THF)] is described. Subsequent hydrolysis produces the corresponding amidine ligands. Furthermore, reaction of the ...
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Papers by Luis Sánchez-Barba