The angel game, as described in [BeCG], has two players the “angel” and the “devil” who play alternately on the 2-dimensional integer lattice, Z. We refer to lattice points as vertices, and write o = (0, 0) for the origin. The angel has a certain fixed power p. (We refer to it as a p-angel .) It starts the game at the origin, and at each play moves to another vertex so that the change in each c...

We construct nonconstructible simplicial d-spheres with d+10 vertices and nonconstructible, nonrealizable simplicial d-balls with d+ 9 vertices for d ≥ 3.

In this paper, we examine non-stretchable 2D polygonal cloth, and place bounds on the number of fingers needed to immobilize it. For any non-stretchable cloth polygon, it is always necessary to pin all the convex vertices. We show that for some shapes, more fingers are necessary. No more than one third of the concave vertices need to be pinned for simple polygons, and no more than one third of ...

A Boolean function is called vertex-transitive, if the partition of the Boolean cube into the preimage of 0 and the preimage of 1 is invariant under a vertex-transitive group of isometric transformations of the Boolean cube. The logarithm of the number of the vertex-transitive functions of n variables is at least Ω(n) and at most O(n logn). There is a polynomial over F2 of any given degree, whi...

We construct the first explicit example of a simplicial 3-ball B15,66 that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B12,38 with 12 vertices that is collapsible and not shellable, but evasive. Finally, we present the first explicit triangulation of a 3-sphere S18,125 (with only 18 vertices) that is not locally constructible. All these examples are based on knotted ...

In this paper, obtain necessary and sufficient condition for 3-connected cubic graph to have a containing any set of nine vertices and an Vve also prove that in every 3-connected cubic planar any set of fourteen vertices and is contained in a cycle. As there is 3-connected cubic planar that of fifteen vertices and an not on any cycle, the result is the best

Let P~,..., Pk be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices. A start vertex, in general, is a vertex both of which neighbors have larger x coordinate. We present an algorithm for triangulating P~,..., Pk in time O(n + s log s). s may be viewed as a measure of non-convexity. In particular, s is always bounded by the number of concave angles + 1, an...

We present a detailed study of the dynamics associated with ghost sector quenched QCD in Landau gauge, where relevant dynamical equations are supplemented key inputs originating from large-volume lattice simulations. In particular, we solve coupled system Schwinger-Dyson that governs evolution dressing function and ghost-gluon vertex, using as input for gluon propagator data have been cured vol...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

Let $D$ be a finite and simple digraph with vertex set $V(D)$‎.‎A signed total Roman $k$-dominating function (STR$k$DF) on‎‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...