Suitable Triangulation Research Articles

Generalization of Heron\u2019s and Brahmagupta\u2019s equalities to any cyclic polygon

It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n>4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

Computation of non-monotonic Lyapunov functions for continuous-time systems

In this paper, we propose two methods to compute non-monotonic Lyapunov functions for continuous-time systems which are asymptotically stable. The first method is to solve a linear optimization problem on a compact and bounded set. The proposed linear programming based algorithm delivers a CPA11continuous and piecewise affine non-monotonic Lyapunov function on a suitable triangulation covering the given compact and bounded set excluding a small neighbourhood of the equilibrium. It is shown that for every asymptotically stable system there exists a suitable triangulation such that the proposed algorithm terminates successfully. The second method is to verify a CPA function constructed based on the values of the norm of the state at all vertices of a suitable triangulation covering the given compact and bounded set is a non-monotonic Lyapunov function on the given set without a small neighbourhood of the equilibrium. It is further proved that if system is asymptotically stable then there exists a suitable triangulation such that the second way works. The comparison of the proposed two methods are discussed via three examples.

Computation of local ISS Lyapunov functions for discrete-time systems via linear programming

This paper presents a numerical algorithm for computing ISS Lyapunov functions for discrete-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on solving a linear optimisation problem and delivers a continuous and piecewise affine ISS Lyapunov function on a suitable triangulation covering the given compact set excluding a small neighbourhood of the origin. The objective of the linear optimisation problem is to minimise the ISS gain. It is shown that for every ISS system there exists a suitable triangulation such that the proposed algorithm terminates successfully.

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.

Envy-free division of discrete cakes

We address the problem of existence of an envy-free distribution of pieces among two or more players in the cake-cutting setting with the minimum number of cuts. Our cakes are discrete in the sense that the players’ valuations are concentrated on atoms. These atoms are placed on an interval and no two players give positive values to atoms placed at the same position. We prove the existence of an envy-free allocation for any discrete cake and any number of players by resorting to Sperner’s Lemma, a suitable triangulation, and moving-knife arguments. Our results also apply to pies, which are defined over circumferences instead of intervals.

Envy-free division of discrete cakes

Abstract We address the problem of existence of an envy-free distribution of pieces among two or more players in the cake-cutting setting with the minimum number of cuts. Our cakes are discrete in the sense that the playersʼ valuations are concentrated on atoms. These atoms are placed on an interval and no two players give positive value to atoms placed at the same position. We prove the existence of an envy-free allocation for any discrete cake and any number of players by resorting to Spernerʼs Lemma, a suitable triangulation, and moving-knife arguments. Our results also apply to pies, which are defined over circumferences instead of intervals.

Neuwirth manifolds and colourings of graphs

The paper uses the fact that PL manifolds may be studied through graphs with coloured edges. The representation is given by taking the l-skeleton of the cellular subdivision dual to a suitable triangulation (minimal with respect to the vertices) of a manifold. Here we describe a very simple construction to obtain a 4-coloured graph representing a closed orientable 3-manifold from its Heegaard diagram. This construction allows us to completely classify a countable class of closed orientable 3-manifoldsMn,n ⩾ 3, introduced by L. Neuwirth in Proc. Camb. Phil. Soc.64 (1968), 603–613. Indeed, we show thatMn is homeomorphic to the Seifert fibered space $$\sum _n = (0 \circ 0 / - 1 \underbrace {(2,1) (2,1) ... (2,1)).}_{n times}$$ It is also proved thatMn is the unique closed 3-manifold having the canonical 2-complex associated to the standard presentation of the Fibonacci groupF(n − 1, n) as its spine.

Adaptive Approximation by Piecewise Linear Polynomials on Triangulations of Subsets of Scattered Data

Given a set V of data points in $R^2 $ with corresponding data values, the problem of adaptive piecewise polynomial approximation is to choose a subset of points of V, to create a triangulation of this subset, and to define a piecewise linear surface over the triangulation such that the deviation of this surface from the data set is no more than a prescribed error tolerance. A typical numerical scheme starts with some initial triangulation and adds more points (and triangles) as necessary until the resulting piecewise linear surface satisfies the error bound. In this paper two ingredients of such schemes are discussed. The first problem is that of constructing a suitable triangulation of a subset of points. The use of data-dependent triangulations that depend on the given function values at the data points is discussed, and some data-dependent criteria for optimizing a triangulation are presented and compared to the Delaunay criterion leading to the well-known Delaunay triangulation traditionally used for this purpose. The second problem addressed in this paper is how to select a piecewise linear surface approximating the given data. A common approach is to use an interpolating surface, i.e., to require that the surface interpolates the data at the nodes of the triangulation. In this paper the least-square approximation to the data from the space of piecewise linear polynomials defined over a triangulation of a subset of V is used. It is proved that the matrix of the normal equations is always nonsingular and a bound for its condition number is derived. This bound is relatively low, hence the least-square surface can be computed by solving the normal equations, e.g., by the conjugate gradient scheme with no preconditioning. Various numerical experiments demonstrate the improvement in the quality of the approximation when certain data-dependent triangulations are used. Improvement is also reported when least-square surfaces are compared to interpolating surfaces.