In this paper, an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever for some s > 0, the solution can be approximated to accuracy O(n−s) in energy norm by a continuous piecewise linear function on some partition with n triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate ...

Let AA A A be a triangle with vertices at A1, A2 and A3. The process of "bisecting AA A A is defined as follows. We first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new triangles, AA'DAi+2 and ADAi+1Ai+2. Let Aqq be a given triangle, with smallest interior angle a > 0. Bisect AQ0 into two new triangles, Aj., í = 1, 2. Next, bisect eac...

We show that every K4-free planar graph with at most ν edge-disjoint triangles contains a set of at most 32ν edges whose removal makes the graph triangle-free. Moreover, equality is attained only when G is the edge-disjoint union of 5-wheels plus possibly some edges that are not in triangles. We also show that the same statement is true if instead of planar graphs we consider the class of graph...

In the paper we present purely combinatorial conditions that allow us to recognize the topological equivalence (or non-equivalence) of two given dissections. Using a computer program based on this result, we are able to generate a set which contains all topologically non-equivalent dissections of a p0-gon into convex pi-gons, i = 1, ..., n, where n, p0, ..., pn are integers such that n ≥ 2, pi ...

In this study, we aim to understand the origins of human intuitions about Euclidean geometry by simulating geometric concepts acquisition with unsupervised learning in hierarchical generative models. Specifically, we build a deep neural network that learns a hierarchical generative model of sensory inputs. The results show that hidden layer activities can support the categorization of different...

We have shown previously that the precision of angle judgments depends strongly on the global stimulus configuration: discrimination thresholds for angles that form part of isosceles triangles are up to 3 times lower than for those that form part of scalene triangles [Kennedy, G. J., Orbach, H. S., & Loffler, G. (2006). Effects of global shape on angle discrimination. VisionResearch, 46(8-9), 1...

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if p ≥ max ( log n t , log n √ t ), where n, t, ∆ denote the number of vertices in G, the number of triangles in G, the maximum number of triangles an edge of G is cont...

Information on su(N) tensor product multiplicities is neatly encoded in Berenstein-Zelevinsky triangles. Here we study a generalisation of these triangles by allowing negative as well as positive integer entries. For a fixed triple product of weights, these generalised Berenstein-Zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so-called virtual...

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to their boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on ...

A visibility algorithm for triangulated panoramas is proposed. The algorithm can correctly resolve the visibility without making use of any depth information. It is especially useful when depth information is not available, such as in the case of real-world photographs. Based on the optical flow information and the image intensity, the panorama is subdivided into variable-sized triangles, image...