We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we that if a complex satisfies Serre's condition $(S_{\ell})$ then so do all of its selected subcomplexes. also provide formula the depth in terms reduced homologies By passing to barycentric subdivision, our results give information about and any complex. Our extend proved by Stanley, Munkres, Hibi.

The non-Hausdorff suspension of the one-sphere S1 of complex numbers fails to model the group’s continuous multiplication. Moreover, finite connected H-spaces are contractible, and therefore cannot model infinite connected non-contractible H-spaces. For an H-space and a finite model of the topology, the multiplication can be realized on the finite model after barycentric subdivision.

For simplicial complexes and maps, the notion of being in same contiguity class is defined as discrete version homotopy. In this paper, we study distance, $SD$, between two maps adapted from homotopic distance. particular, show that versions $LS$-category topological complexity are particular cases more general notion. Moreover, present behaviour $SD$ under barycentric subdivision, its relation...

2D Simulation and Mapping using the Cauchy-Green Complex Barycentric Coordinates Conformal maps are especially useful in geometry processing for computing shape preserving deformations, image warping and manipulating harmonic functions. The Cauchy-Green coordinates are complex-valued barycentric coordinates, which can be used to parameterize a space of conformal maps from a planar domain bounde...

We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in 2D. Our method extends weighted Gordon-Wixom interpolation to non-convex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with smooth boundaries. We achieve these properties by using the distance to lines tangent to the boun...

In this paper we discuss a natural way to deene barycentric coordinates on general sphere-like surfaces. This leads to a theory of Bernstein-B ezier polynomials which parallels the familiar planar case. Our constructions are based on a study of homogeneous polynomials on trihedra in IR 3. The special case of Bernstein-B ezier polynomials on a sphere is considered in detail.

We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are oneto-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can...

Let G be a finite Abelian group of order n. A k-sequence in G is said to be barycentric if it contains an element which is the “average” of its terms. The kbarycentric Olson constant BO(k, G) is introduced as the minimal positive integer t such that any t-set in G contains a k-barycentric set. Conditions for the existence of BO(k, G) are established and some values or bounds are given. Moreover...

Image morphing, which is a 2D imaging technique, allows smooth transition between images. However, a limitation of existing image morphing techniques is the lack of user interaction. Another limitation is that shape warping often causes distortion due to barycentric mapping. In this paper, we present a novel 3D morphological technique to address these problems. A new concept of relief occlusion...

S d := {x = (x1, . . . , xd) ∈ R : 0 6 x1, . . . , xd 6 1, 0 6 x1 + · · ·+ xd 6 1} denote the standard simplex in R. We denote by ∂S the boundary of S. We will also use barycentric coordinates on the simplex which we denote by the boldface symbol x = (x0, x1, . . . , xd), x0 := 1−x1−· · ·−xd. We will use standard multiindex notation such as x := x0 0 x α1 1 · · ·xd d and α n := (α0 n , α1 n , ·...