Cluster of clusters: Structural rules for vertex-sharing icosahedral supraclusters

1990 ◽  
Vol 1 (2) ◽  
pp. 223-228 ◽  
Author(s):  
Boon K. Teo ◽  
Hong Zhang
Keyword(s):  
Structural Rules ◽  
Vertex Sharing ◽  
Cluster Of Clusters

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

scholarly journals The crystal structure of szenicsite, Cu3MoO4(OH)4

1998 ◽  
Vol 62 (04) ◽  
pp. 461-469 ◽  
Author(s):  
Peter C. Burns
Keyword(s):  
Crystal Structure ◽  
Goodness Of Fit ◽  
Direct Methods ◽  
Ligand Field ◽  
Area Detector ◽  
Vertex Sharing ◽  
Distorted Octahedral ◽  
Jahn Teller ◽  
Jahn Teller Effect ◽  
Least Squares Techniques

Abstract The crystal structure of szenicsite, Cu3MoO4(OH)4, orthorhombic, a = 8.5201(8), b = 12.545(1), c = 6.0794(6) Å, V = 649.8(2) Å3, space group Pnnm, Z = 4, has been solved by direct methods and refined by least-squares techniques to an agreement index (R) of 3.34% and a goodness-of-fit (S) of 1.11 for 686 unique observed [|F| ⩾ 4σF] reflections collected using graphite-monochromated Mo-Kα X-radiation and a CCD area detector. The structure contains three unique Cu2+ positions that are each coordinated by six anions in distorted octahedral arrangements; the distortions of the octahedra are due to the Jahn-Teller effect associated with a d 9 metal in an octahedral ligand-field. The single unique Mo6+ position is tetrahedrally coordinated by four O2− anions. The Cu2+ϕ6 (ϕ: unspecified ligand) octahedra share trans edges to form rutile-like chains, three of which join by the sharing of octahedral edges to form triple chains that are parallel to [001]. The MoO4 tetrahedra are linked to either side of the triple chain of Cu2+ϕ6 octahedra by the sharing of two vertices per tetrahedron, and the resulting chains are cross-linked through tetrahedral-octahedral vertex sharing to form a framework structure. The structure of szenicsite is closely related to that of antlerite, Cu3SO4(OH)4, which contains similar triple chains of edge-sharing Cu2+ϕ6 octahedra.

On-Line Chinese Character Recognition Via a Representation of Spatial Relationships Between Strokes

1997 ◽  
Vol 11 (03) ◽  
pp. 329-357 ◽  
Author(s):  
Ju-Wei Chen ◽  
Suh-Yin Lee
Keyword(s):  
Character Recognition ◽  
Chinese Character ◽  
Spatial Relationship ◽  
Spatial Relations ◽  
Chinese Characters ◽  
Spatial Relationships ◽  
Matching Problem ◽  
Structural Rules ◽  
Chinese Character Recognition ◽  
On Line

Chinese characters are constructed by basic strokes based on structural rules. In handwritten characters, the shapes of the strokes may vary to some extent, but the spatial relations and geometric configurations of the strokes are usually maintained. Therefore these spatial relations and configurations could be regarded as invariant features and could be used in the recognition of handwritten Chinese characters. In this paper, we investigate the structural knowledge in Chinese characters and propose the stroke spatial relationship representation (SSRR) to describe Chinese characters. An On-Line Chinese Character Recognition (OLCCR) method using the SSRR is also presented. With SSRR, each character is processed and is represented by an attribute graph. The process of character recognition is thereby transformed into a graph matching problem. After careful analysis, the basic spatial relationship between strokes can be characterized into five classes. A bitwise representation is adopted in the design of the data structure to reduce storage requirements and to speed up character matching. The strategy of hierarchical search in the preclassification improves the recognition speed. Basically, the attribute graph model is a generalized character representation that provides a useful and convenient representation for newly added characters in an OLCCR system with automatic learning capability. The significance of the structural approach of character recognition using spatial relationships is analyzed and is proved by experiments. Realistic testing is provided to show the effectiveness of the proposed method.

Structural Interface for Automated Compliance Checking

2021 ◽  
Author(s):  
Erik Arthur Bjorkner
Keyword(s):  
Cost Reduction ◽  
Design Model ◽  
3D Design ◽  
Structural Rules ◽  
Structural Interface ◽  
Compliance Checking ◽  
Contract Specification

This paper describes the benefits of cost reduction and improved schedule attainment by adding digitized regulatory structural rules and contract specification requirements to the 3D design model through Knowledge Provisioning.

Sign in / Sign up

Export Citation Format

Share Document