Related papers: The Pascal Triangle of a Discrete Image: Definitio…
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups.…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…
The depth of a visible surface of a scene is the distance between the surface and the sensor. Recovering depth information from two-dimensional images of a scene is an important task in computer vision that can assist numerous applications…
Our work is concerned with simplicial complexes that describe higher-order interactions in real complex systems. This description allows to go beyond the pairwise node-to-node representation that simple networks provide and to capture a…
We generalize the concept of Pascal matrices to matrices associated with sets of points by considering multidimensional binomial coefficients as entries. We study their properties and prove that the infinite matrix associated with the set…
In this paper we consider a quantum version of Pascal's triangle. Pascal's triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first prove…
The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the…
The discrete cosine transform is a valuable tool in analysis of data on undirected rectangular grids, like images. In this paper it is shown how one can define an analogue of the discrete cosine transform on triangles. This is done by…
Let $\Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $\Gamma$ is discrete. Historically discreteness algorithms have been…
Image moments are weighted sums over pixel values in a given image and are used in object detection and localization. Raw image moments are derived directly from the image and are fundamental in deriving moment invariants quantities. The…
We continue to investigate combinatorial properties of functions $f_m$ and $c_m$ considered in our previous papers. They depend on an initial arithmetic function $f_0$. In this paper, the values of $f_0$ are the binomial coefficients. We…
A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if…
A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is…
The special Galilean group, usually denoted SGal(3), is a 10-dimensional Lie group whose important subgroups include the special orthogonal group, the special Euclidean group, and the group of extended poses. We briefly describe SGal(3) and…
Stack triangulations appear as natural objects when defining an increasing family of triangulations by successive additions of vertices. We consider two different probability distributions for such objects. We represent, or "draw" these…
Image triangulation, the practice of decomposing images into triangles, deliberately employs simplification to create an abstracted representation. While triangulating an image is a relatively simple process, difficulties arise when…
We define and characterize the $\gamma$-matrix associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. We also define and characterize the $\gamma$-matrix of the reversions of these triangles, in the…
A discrete system constituted of particles interacting by means of a centroid-based law is numerically investigated. The elements of the system move in the plane, and the range of the interaction can be varied from a more local form…
In the context of A. Connes' spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes' construction of spectral triples for group algebras…