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This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

We introduce new techniques for working with presentations for a large class of (strict) tensor categories. We then apply the general theory to obtain presentations for partition, Brauer and Temperley–Lieb categories, as well as several categories of (partial) braids, vines, and mappings.

To every group G we associate a linear monoidal category Par(G) that we call a group partition category . We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of Par(G) into the group Heisenberg category associated to G . This embedding intertwines the natural actions of both categories on modules for wreath products of G . Finally, we prove that the additive Karoubi envelope of Par(G) is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.

The aim of testing based fault localization （TBFL） involves improving the efficiency of program debugging by providing developers with a guide of ranked list of suspicious statements. However, collection of testing information of the whole original test-suite is excessively expensive or even infeasible for developers to conduct TBFL. Traditional test-suite reduction （TSR） techniques are utilized to reduce the size of test- suite. However, they entail a time-consuming process of whole testing information collection. In this study, the distance based test-suite reduction （DTSR） technique is proposed. As opposed to the whole testing information, the distances among the test cases are used to guide the process of test-suite reduction in DTSR. Hence, it is only necessary to collect the testing information for a portion of the test cases for TSR and TBFL. The investigation on the Siemens and SIR benchmarks reveals that DTSR can effectively reduce the size of the given test-suite as well as the time cost of TBFL. Additionally, the fault locating effectiveness of DTSR results is close to that when the whole test-suite is used.

This work describes an empirical investigation of the cost effectiveness of well-known state-based testing techniques for classes or clusters of classes that exhibit a state-dependent behavior. This is practically relevant as many object-oriented methodologies recommend modeling such components with statecharts which can then be used as a basis for testing. Our results, based on a series of three experiments, show that in most cases state-based techniques are not likely to be sufficient by themselves to catch most of the faults present in the code. Though useful, they need to be complemented with black-box, functional testing. We focus here on a particular technique, Category Partition, as this is the most commonly used and referenced black-box, functional testing technique. Two different oracle strategies have been applied for checking the success of test cases. One is a very precise oracle checking the concrete state of objects whereas the other one is based on the notion of state invariant (abstract states). Results show that there is a significant difference between them, both in terms of fault detection and cost. This is therefore an important choice to make that should be driven by the characteristics of the component to be tested, such as its criticality, complexity, and test budget.

The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found.

We describe in this paper a choice relation framework for supporting category-partition test case generation. We capture the constraints among various values (or ranges of values) of the parameters and environment conditions identified from the specification, known formally as choices. We express these constraints in terms of relations among choices and combinations of choices, known formally as test frames. We propose a theoretical backbone and techniques for consistency checks and automatic deductions of relations. Based on the theory, algorithms have been developed for generating test frames from the relations. These test frames can then be used as the basis for generating test cases. Our algorithms take into consideration the resource constraints specified by software testers, thus maintaining the effectiveness of the test frames (and hence test cases) generated.

This chapter contains sections titled: Introduction Impact of Object-Oriented Design on Testing Specification-Based Testing Techniques UML Intraclass Testing UML Interclass Testing Algebraic Testing Techniques Code-Based Testing Techniques Intraclass Structural Testing Interclass Structural Testing Testing in the Presence of Inheritance Regression Testing Conclusions References

A theory of fuzzy objects is derived in the category SpaceFP of spaces with fuzzy partitions, which generalize classical fuzzy sets and extensional maps in sets with similarity relations. It is proved that fuzzy objects in SpaceFP can be characterized by some morphisms in the category of sets with similarity relations. A powerset object functor F in the category SpaceFP is introduced and it is proved that F defines a CSLAT -powerset theory in the sense of Rodabaugh.

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.