Proper Faces Research Articles

Analytical railway line capacity methods from graph-theoretical and polyhedral perspectives

ABSTRACT This paper rigorously addresses the consumed capacity underestimation caused by division-based methods in graph theory and polyhedral geometry. Graph-theoretically, such underestimation occurs because subgraphs in shorter sections approximate their combined graph. Our polyhedral study suggests that the capacity analysis is computationally ‘easy’ because its linear programme in the complete section has an orthogonal projection on the Euclidean plane containing five proper faces, one being the objective-defined one-dimensional optimal facet. Geometrically, we conclude that the polyhedra of the linear programmes in shorter and complete sections have parallel facets, with the former nested within the latter. Consequently, the optimal values of the former cannot exceed those of the latter. With these theoretical insights, we derive a train operation labelling algorithm that takes proven linear time. Beyond enabling faster capacity computation, the findings deepen our understanding of the connection between infrastructure capacity utilisation and timetable structure, ultimately leading to more efficient railway optimisation.

Cohomology rings of a class of torus manifolds

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of locally standard torus manifolds whose orbit space may have proper non-acyclic faces.

A Solution to the Faceless Problem

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

Local super antimagic total face coloring of planar graphs

We using graph G = (V(G), E(G), F(G)) be a nontrivial, finite, connected graph, and a g bijective function mapping total labeling of graph to natural number start form 1 until the sum of vertices, edge, and faces. The sum of vertices, edges, and faces labels in a face f is called the weight of the face f ∈ F(G). If any adjacent two faces f1 and f2 have different weights w(f1) ≠ w(f2) for f1, f2 ∈ F(G), then g is called a labeling of local antimagic total face. We call labeling of local antimagic total face is super if we add vertices label start from 1 until the sum of vertices, edges label start from the sum of vertices plus 1 until the sum of vertices and edges, and faces label start form the sum of vertices and edges plus one untul the sum of vertices, edges, and faces. The local super antimagic total face labeling that induces a proper faces coloring of G where the faces f is assigned by the color w(f) is called local super antimagic total face coloring. The minimum number of colors in local super antimagic total face coloring is local antimagic total face chromatic number and denoted by γlatf(G). In this paper, we used some planar graph such as wheel graph (Wn), jahangir graph (J(2, n)), ladder graph (Ln), and circular ladder graph (CLn). Our results attained the lower bound of local super antimagic total face chromatic number.

Locally standard torus actions and sheaves over Buchsbaum posets

Manifolds with locally standard half-dimensional torus actions represent a large and important class of spaces. Cohomology rings of such manifolds are known in particular cases, but in general even Betti numbers are difficult to compute. Our approach to this problem is the following: we consider the orbit type filtration on a manifold with locally standard action and study the induced spectral sequence in homology. It collapses at the second page only in the case when the orbit space is homologically trivial. The cohomology ring in this case has already been computed. Nevertheless, we can completely describe the spectral sequence under more general assumptions, namely when all proper faces of the orbit space are acyclic. The theory of sheaves and cosheaves on finite partially ordered sets is used in the computation. We establish generalizations of the Poincare duality and the Zeeman-McCrory spectral sequence for sheaves of ideals of exterior algebras. Bibliography: 15 titles.

Tingley's problem on finite von Neumann algebras

In this paper, we give an affirmative answer to Tingley's problem between finite von Neumann algebras. To do this, we see that every norm closed proper face of the unit ball of a unital C⁎-algebra can be represented as the intersection of a family of maximal proper faces.

Cohomology of toric origami manifolds with acyclic proper faces

Toric origami manifolds are generalizations of symplectic toric manifolds, where the origami symplectic form, in contrast to the usual sym- plectic form, is allowed to degenerate in a good controllable way. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes. The cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. One can obtain a similar description for the cohomology of a toric origami manifold M in terms of the orbit space M=T when M is orientable and the orbit space M=T is con- tractible; this was done by Holm and Pires in (9). Generally, the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that any proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus action in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.

Locally standard torus actions and $h'$-numbers of simplicial posets

We consider the orbit type filtration on a manifold with a locally standard torus action and study the corresponding spectral sequence in homology. When all proper faces of the orbit space are acyclic and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms of its second page are equal to $h'$-numbers of a simplicial poset dual to the orbit space. Betti numbers of a manifold with a locally standard torus action are computed: they depend on the combinatorics and topology of the orbit space but not on the characteristic function. A toric space whose orbit space is the cone over a Buchsbaum simplicial poset is studied by the same homological method. In this case the ranks of the diagonal terms of the spectral sequence at infinity are the $h''$-numbers of the simplicial poset. This fact provides a topological evidence for the nonnegativity of $h''$-numbers of Buchsbaum simplicial posets and links toric topology to some recent developments in enumerative combinatorics.

Boundary of the set of separable states

Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S 1 , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . To any subspace V ⊆ H , we associate a face F V of S 1 consisting of all states ρ ∈ S 1 whose range is contained in V . We prove that F V is a maximal face if and only if V is a hyperplane. If V =| ψ 〉 ⊥ , where | ψ 〉 is a product vector, we prove that Dim F V = d 2 − 1 − ∏ ( 2 d i − 1 ) , where d i = Dim H i and d = ∏ d i . We classify the maximal faces of S 1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂ S 1 , of S 1 is the union of all maximal faces. When d &gt;6, it is easy to show that there exist full states on ∂ S 1 , i.e. states ρ ∈ ∂ S 1 such that all partial transposes of ρ (including ρ itself) have rank d . Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b &gt;0, b ≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b = 0 , 1 , ∞ .

Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm

Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion–exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using partially open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space.

On zonoids whose polars are zonoids

Zonoids whose polars are zonoids cannot have proper faces other than vertices or facets. However, there exist non-smooth zonoids whose polars are zonoids. Examples in ℝ and ℝ are given.

Projections of polytopes and the generalized baues conjecture

Associated with every projection ź:Pźź(P) of a polytopeP is a partially ordered set of all locally coherent strings: the families of proper faces ofP that project to valid subdivisions of ź(P), partially ordered by the natural inclusion relation. The Generalized Baues Conjecture posed by Billeraet al. [4] asked whether this partially ordered set always has the homotopy type of a sphere of dimension dim(P--dim(ź(P))ź1. We show that this is true in the cases when dim(ź(P))=1 (see[4]) and when dim(P)--dim(ź(P))≤2, but fails in general. For an explicit counterexample we produce a nondegenerate projection of a five-dimensional, simplicial, 2-neighborly polytopeP with 10 vertices and 42 facets to a hexagon ź(P)⊆ź2. The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitrary projection of polytopes.

Normal Semimodules: A Theory of Generalized Convex Cones

In [3], C. Davis showed that if a convex polyhedral cone C (the positive span of a finite set of vectors in Euclidean space) contains no nonzero linear subspace, then C is linearly isomorphic to the set V+ of nonnegative points in a linear subspace V of Rn. Moreover n can be taken to be the number of facets (maximal proper faces) of C.

On the compactness of the hyperspace of faces

By a flat we mean a translate of a subspace of R and by a hyperplane, a flat of dimension d — 1. If X is a compact convex subset of R the symbols dim (X), rel int (X), and rel bd (X) denote respectively, the dimension of the flat generated by X, the interior, and the boundary of X with respect to the flat X generates. A hyperplane H is called a supporting hyperplane of X if H Π X Φ 0 and H n relint (X) = 0 . A set A is called a face of X if A = X, A = 0 or if there exists a supporting hyperplane H of X such that i = J ϊ n X The set of proper faces (those not X or 0) is denoted by ^(X). A set J3 is called a poonem of X if there exists sets j?o, Blf , £ m such that £ m = X and B^eJ^iB,) for i = 1, , m. The set of poonems of X is denoted by &*(X). A point x in X is called a ^-exposed [A -extreme] point if for some j <^k, x belongs to a i-dimensional face [i-dimensional poonem] of X. The symbols expfe (X) and ext^ (X) denote the set of fc-exposed and fc-extreme points of X respectively. A point x of X is called an exposed point of X if {x} G ^(X) and x is called an extreme point if whenever x 6 [a, b] c X, we have x = a or x = b, where [a, b] denotes the closed line segment from a to 6. The symbols ext (X) and exp (X) denote the extreme points and exposed points of X, respectively. Note that exp (X) — exp0 (X) and ext (X) = ext0 (X). Also, let q denote the Hausdorff metric. Finally, if DaR, the symbols CI(JD) and conv (D) denote the closure of D and the convex hull of D respectively. We require the following results

Homology cycles in manifolds with locally standard torus actions

Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathbb{Z}[S_Q]/\Theta)/W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset $S_Q$ dual to $Q$; $\Theta$ is the linear system of parameters determined by the characteristic function; and $W$ is a certain submodule, lying in the socle of $\mathbb{Z}[S_Q]/\Theta$. Intersections of homology classes different from face submanifolds are described in terms of intersections on $Q$ and $T^n$.