Correction to: MPS Stability and the Intersection Property

Topological properties of objects should be maintained and preserved to concisely represent objects. However, the implementation of 2D topological rules requires the decomposition of 3D objects into lower dimensions to determine topological relationships. This results in 2D topological relationships although the connected objects are in 3D. Hence, accurate representation of 3D connectivity in 3D models is limited. 3D topological rules can be implemented to include topological connectivity in 3D space. This paper implemented an extension of the 27-Intersection Model (27-IM) called the 36-Intersection Model (36-IM) to represent 3D topological adjacencies of two objects in 3D space. This resulted in a 12 × 3 intersection matrix or 36-IM that represented the intersections in terms of dimension and number of separations. Six cases were tested, consisting of “meets”, “disjoint” “intersects at a line”, “intersects at a point”, “contains”, and “overlaps”. The resulting 36-IM matrices provided an accurate representation of how the objects in 3D space were related and their dimension of intersections. The formalisms of the 36-IM matrices were also interoperable which allowed the interpretation of 36-IM using the 9IM and DE-9IM to determine general topological relationships. By examining the intersection of interiors, boundaries and exteriors of 3D objects without object decomposition, 3D topological relationships can be determined as well as the dimension and manner of intersection.

Let V \mathcal {V} (resp. V ∗ {\mathcal {V}^*} ) be the set of all closed, convex and bounded (resp. w ∗ {w^*} -compact and convex) subsets of a Banach space E E (resp. of its dual E ∗ {E^*} ) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm | | ⋅ | | || \cdot || in E E with dual | | ⋅ | | ∗ || \cdot |{|^*} such that ( E , | | ⋅ | | ) \left ( {E,|| \cdot ||} \right ) has Mazur’s intersection property and ( E ∗ , | | ⋅ | | ∗ ) \left ( {{E^*},||\cdot |{|^*}} \right ) has w ∗ {w^*} -Mazur’s intersection property, then (1) there exists a dense G δ {G_\delta } subset V 0 {\mathcal {V}_0} of V \mathcal {V} such that for every X ∈ V 0 X \in {\mathcal {V}_0} the strongly exposing functionals form a dense G δ {G_\delta } subset of E ∗ {E^*} ; (2) there exists a dense G δ {G_\delta } subset V 0 ∗ \mathcal {V}_0^* of V ∗ {\mathcal {V}^*} such that for every X ∗ ∈ V 0 ∗ {X^*} \in \mathcal {V}_0^* the w ∗ {w^*} -strongly exposing functionals form a dense G δ {G_\delta } subset of E E . In particular every X ∈ V 0 X \in {\mathcal {V}_0} is the closed convex hull of its strongly exposed points and every X ∗ ∈ V 0 ∗ {X^*} \in \mathcal {V}_0^* is the w ∗ {w^*} -closed convex hull of its w ∗ {w^*} -strongly exposed points.

We identify a property of the local tensors of matrix product states (MPS) that guarantees that their parent Hamiltonians satisfy the intersection property. The intersection property ensures that the ground space consists of MPS, with degeneracy bounded by the square of the bond dimension. The new local property, dubbed stability, generalizes (block) injectivity and is satisfied by the MPS tensors that construct the W state, domain wall superposition states, and their generalizations.

Intersection properties of the unit ball

The Southern Jiangxi Province (SJP) hosts one of the best known districts of tungsten deposits in the world. Delineating spatial complexities of geological features and their controls on regional-scale tungsten mineralization by using an integrated fractal and weights-of-evidence (WofE) method can provide insights into the understanding of ore genesis and facilitate further prospecting in this area. The box-counting fractal analysis shows that most of the tungsten occurrences are distributed in regions with high fractal dimensions of faults and fault intersections, suggesting ore-forming favorability of areas with highly complex structural patterns. The WofE-derived indices are employed to quantitatively measure the controls of analyzed features on mineralization, which illustrate that tungsten anomalies, faults, Yanshanian granites, and manganese anomalies have high contrast values, implying a spatially strong correlation of these features with tungsten occurrences. In particular, high manganese anomalies in host rock may provide a novel indication for mineral prospecting in this area. A predictive map is extracted based on the combination of fractal and WofE results, providing intuitive guides for future prospectivity in this area. Regions identified by high posterior probability in conjunction with high fractal dimensions of both faults and fault intersections are evaluated as the most favorable targets.

We obtain an estimate for the typical Hausdorff dimension of the intersection of a set $E$ with homothetic copies of a set $F$, where $E$ and $F$ are Borel subsets of $\mathbb{R}^{n}$. We apply this to the `distance set problem' for a polyhedral norm on $\mathbb{R}^{n}$, by showing that there are subsets of full dimension with distance set of Lebesgue measure 0.

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.

In this paper we used the concept of a pure submodule relative to submodule T in two concepts, pure relative to submodule T Baer injective modules and module with pure relative to submodule T intersection property. Some properties and some characterization of this notions are established.

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

Protocols to solve several distributed issues, such as name service, mutual exclusion, and creation of an atomic shared register, require two types of subsets with intersection property. Distributed match-making provides a method of creating the subsets, and the lower bound of the number of messages to solve the issues. This paper discusses the fault-tolerant and weighted case, in which a protocol is fault-tolerant regarding node failures, and in which weights of subsets are different. The paper first provides the lower bound of the number of messages required for a protocol in a general form. Then, it concentrates a symmetric case and shows the lower bound in a simpler form. The paper also provides a method of constructing the two types of subsets, which realize the lower bound. It first shows a method for a fully symmetric case, and extends it for other cases. The extended method is practical. It creates a cyclic communication structure, and is valid for any degree of fault-tolerance and weights.

A fundamental tool to prove inner bounds in classical network information theory is the so-called ‘conditional joint typicality lemma’. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments do not work in the quantum setting. This bottleneck shows up in the fact that so-called ‘simultaneous decoders’, as opposed to ‘successive cancellation decoders’, are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called ‘simultaneous smoothing’ theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so we develop two novel tools in quantum information theory, which may be of independent interest. The first tool is a simple geometric idea called tilting, which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing and augmentation, is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski's theorem on convex polytopes. Also we show that for any Cohen-Macaulay cell complex as above, although there is no generalization of the Stanley-Reisner ring of simplicial complexes, there is a generalization of its canonical module.

On the weak ∗ Mazur intersection property and Frechet differentiable norms on dense open sets

The purpose of this note is to demonstrate that all Kleinian groups have the finitely generated intersection property; that is, the intersection of any pair of fnitely generated subgroups of 2 Klcinian group is again finitely generated. This rcsuit is a generalization of a theorem of Hempel [I], which states that geometrically finite Kleinian groups have the Cinirely gencratcd intersection property.

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that no two sets $A, B \in \mathcal{F}$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.