Kenji Ohmori

Designing and modeling cyberworlds using the incrementally modular abstraction hierarchy based on homotopy theory

For designing and modeling complicated and sophisticated systems such as cyberworlds, their mathematical foundation is critical. To realize it, two important properties called the homotopy lifting property (HLP) and homotopy extension property (HEP) are applied for designing and modeling a system in a bottom-up way and a top-down way, respectively. In this paper, an enterprise system and a real-time embedded system are considered as important socially emerging cases of cyberworlds, where the π-calculus processes for describing these behaviors formally, a Petri net for explaining process interactions, and XMOS XC programs are modeled and designed by our approach. The spaces in both properties are specified by the incrementally modular abstraction hierarchy by climbing down the abstraction hierarchy from the most abstract homotopy level to the most specific view level, while keeping invariants such as homotopy equivalence and topological equivalence.

Shape modeling using homotopy

Introduces a new method of shape modeling using homotopy and object-oriented modeling. Homotopy is a kind of topology that gives more general ideas of presenting invariant properties of geometrical objects and is further expanded to conceptual objects. The conventional shape modeling method, using polygonalization, has serious difficulties in preserving invariant properties, leading to the necessity of a massive amount of data. On the other hand, the combination of homotopy and object-oriented modeling, which uses class hierarchy, helps to preserve invariant properties at all abstraction levels. We explain how our new method helps us to preserve invariant properties, which keeps the amount of data to the minimum possible level, using an example of a tennis ball rolling on a slope.

Cyberworlds: architecture and modeling by an incrementally modular abstraction hierarchy

An incrementally modular abstraction hierarchy, which specifies, models and visualizes the architecture of cyberworlds from general to specific, is presented. The hierarchy, consisting of a homotopy level with fiber bundles, a set theoretical space level, a topological space level, an adjunction space level, a cellular space level, and presentation and view-levels, is described theoretically with examples of online book shopping such as e-commerce, seat assembling as such e-manufacturing, and accounting such as e-economy. Sharing invariants defined at each level contributes to robust architecture and modeling for designing, analyzing, implementing and visualizing cyberworlds, which results in a fault-free reduction of time and cost.

A kaleidoscope as a cyberworld and its animation: linear architecture and modeling based on an incrementally modular abstraction hierarchy

An incrementally modular abstraction hierarchy is known to effectively linearize cyberworlds and virtual worlds, which are combinatorially exploding and hardly managed. It climbs down from general level to specific model preserving the higher level modules as invariants. It not only prevents the combinatorial explosion but also benefits the reuse, development, testing and validation of cyberworld resources. By applying this incrementally modular abstraction hierarchy to a kaleidoscope animation, its architecture and modeling is also specified in this paper as a typical case of cyberworlds. In particular, a homotopy lifting property and a homotopy extension property, which satisfy a duality relation, are also described to show how a kaleidoscope world is systematically created top‐down from the whole system and bottom‐up from the components. Copyright

An Incrementally Modular Abstraction Hierarchy for Linear Software Development Methodology

The incrementally modular abstraction hierarchy, which specifies architecture and modeling of cyberworlds and defines the most general invariants at the higher levels and preserving them at the lower levels, are applied to software development to show that its linear increment of modules prevents combinatorial explosion of design- and test-processes

The Mathematical Structure of Cyberworlds

The mathematical structure of cyberworlds is clarified based on the duality of homology lifting property and homotopy extension property. The duality gives bottom-up and top-down methods to model, design and analyze the structure of cyberworlds. The set of homepages representing a cyberworld is transformed into a state finite machine. In development of the cyberworld, a sequence of finite state machines is obtained. This sequence has homotopic property. This property is clarified to map a finite state machine to a simplicial complex. Wikipedia, bottom-up network construction and top-down network analysis are described as examples.Visual clustering should be one of the basic tools for time-dependent data analysis in cyberworlds. This paper describes a novel approach to spatial clustering and boundary detection based on geometric modeling and visualization. Datasets and boundaries of clusters are visualized as 3D points and surfaces of reconstructed solids changing over time. Our approach applies the concepts of geometric solid modeling and uses density as clustering criteria that comes from traditional density-based clustering techniques. Visual clustering allows the user to analyze results of clustering the data changing over time and to interactively choose appropriate parameters.

Visualized Deformation of Joinery to Understand Jointing Process by Homotopy Theory and Attaching Maps

Visualization is effective to understand complicated and sophisticated phenomena. Joiner, which joints pieces of woods and has complex process of jointing, is analyzed and modeled for visualization using homotopy theory and attaching maps. Joining the two pieces of woods changes their structures differently, which depends on jointing techniques. The paper describes difference between the tenon-mortise and lap joints in the fundamental group of homotopy. As the properties of the fundamental group are invariants, they are preserved through modeling from the abstract level to the specific one. When describing the process of jointing, two piece of woods are indirectly connected through a virtual attaching space using an attaching map, which avoids unnecessary consideration of piece relations. Using these concepts of homotopy, the process of jointing two pieces of woods are modeled from the abstract level in homotopy to the middle level of cellular structured spaces, to the specific one of physical presentation.

A Formal Methodology for Developing Enterprise Systems Procedurally: Homotopy, Pi-Calculus and Event-Driven Programs

A new approach for designing and modeling enterprise systems is described. The homotopy lifting property (HLP) is used to design an enterprise system in a bottom up way. As an example, task changes in a department are designed and implemented by our approach: the incrementally modular abstraction hierarchy (IMAH) starting at the most abstract level of homotopy and ending at the most concrete level of program codes. At first, the HLP is constructed as the most abstract level. Then, task changes and a state transition diagram, which constitute of two spaces of the HLP are defined from an abstract level to a concrete level. Then, agent diagrams are obtained in a bottom-up way. The agent diagrams are transformed from an abstract level to a concrete level until program codes written by the C-like programing language are implemented on an event-driven and multi-thread processor XMOS. While carrying out these procedures, invariants are preserved to avoid unnecessary testing, which usually consumes a large amount of time and cost in the traditional approaches. The established method is also effective in modeling an designing cyber worlds.

On-line handwritten Kanji character recognition using hypothesis generation in the space of hierarchical knowledge

For online handwritten Kanji recognition, a new approach that cyclically generates a more concrete hypothesis from the current hypothesis by using hierarchically represented knowledge and that has achieved high recognition rate in reduced processing time is described. The properties of a characteristic stroke are represented in the form of fuzzy rules. Characteristic strokes, each of which is close to part of an input pattern stroke, are found by fuzzy inference, where each input pattern stroke is investigated by using fuzzy rulers.<<ETX>>

Mathematical Foundations for Designing a 3-Dimensional Sketch Book

This paper describes mathematical foundations for designing a 3-dimensional sketch book and the development of an experimental system. The system helps a non-professional user draw a mountain from a rough image to a fine image using filtration. The user draws important critical points (summits, ravine bottoms and saddles), which are used to obtain a 0-dimensional approximation. The system generates a Reeb graph to provide height information of the saddles with a partial order in height. Then, the contours are provided from the Reeb graph. The user draws ridges and ravines by pushing and pulling contours. The system generates a 1-dimensional approximation from these curves. Finally, NURBS surfaces are generated to give a 2-dimensional approximation and a 3-dimensional rendering image is obtained. The above procedure is repeated until a satisfactory result is obtained by giving the most important points and curves at the first stage and adds less important points and curves, later.