Parametric dependence of irregular conformal block

Irregular conformal block is an important tool to study a new type of conformal theories, which can be constructed as the colliding limit of the regular conformal block. The irregular conformal block is realized as the $\beta$-deformed Penner matrix model whose partition function is regarded as the inner product of the irregular modules. The parameter dependence of the inner product is obtained explicitly using the loop equation with close attention to singularities in the parameter space. It is noted that the exact singular structure of the parameter space in general can be found using a very simple and powerful method which uses the flow equations of the partition function together with the hierarchical structure of the singularity. This method gives the exact expression to all orders of large $N$ expansion without using the explicit contour integral of the filling fraction.Comment: 34pages, 8figure

Consistency of a system of equations: What does that mean?

The concept of (structural) consistency also called structural solvability is an important basic tool for analyzing the structure of systems of equations. Our aim is to provide a sound and practically relevant meaning to this concept. The implications of consistency are expressed in terms of explicit density and stability results. We also illustrate, by typical examples, the limitations of the concept

Architecture Diagrams: A Graphical Language for Architecture Style Specification

Architecture styles characterise families of architectures sharing common characteristics. We have recently proposed configuration logics for architecture style specification. In this paper, we study a graphical notation to enhance readability and easiness of expression. We study simple architecture diagrams and a more expressive extension, interval architecture diagrams. For each type of diagrams, we present its semantics, a set of necessary and sufficient consistency conditions and a method that allows to characterise compositionally the specified architectures. We provide several examples illustrating the application of the results. We also present a polynomial-time algorithm for checking that a given architecture conforms to the architecture style specified by a diagram.Comment: In Proceedings ICE 2016, arXiv:1608.0313