Recursive Method for Nekrasov partition function for classical Lie groups

Nekrasov partition function for the supersymmetric gauge theories with general Lie groups is not so far known in a closed form while there is a definition in terms of the integral. In this paper, as an intermediate step to derive it, we give a recursion formula among partition functions, which can be derived from the integral. We apply the method to a toy model which reflects the basic structure of partition functions for BCD type Lie groups and obtained a closed expression for the factor associated with the generalized Young diagram.Comment: 21 pages;v2 comments and references adde

Recent Developments in Copyright Protection for Computer Software in the United States and Japan

Many current legal issues pertaining to copyright of computer software involve defining the scope of protection of non-literal expression, such as user interface and look and feel, in contrast to literal expression, such as source code, to which ownership may be more clearly attributed. Superficially, it appears that the case law pertaining to non-literal expression is developing differently in Japan and the United States. This comment demonstrates that, however, while Japanese and U.S. courts have been applying formally different analytical criteria, the decisions of both have been similar in seeking equity-oriented solutions

Population–reaction model and microbial experimental ecosystems for understanding hierarchical dynamics of ecosystems

Understanding ecosystem dynamics is crucial as contemporary human societies face ecosystem degradation. One of the challenges that needs to be recognized is the complex hierarchical dynamics. Conventional dynamic models in ecology often represent only the population level and have yet to include the dynamics of the sub-organism level, which makes an ecosystem a complex adaptive system that shows characteristic behaviors such as resilience and regime shifts. The neglect of the sub-organism level in the conventional dynamic models would be because integrating multiple hierarchical levels makes the models unnecessarily complex unless supporting experimental data are present. Now that large amounts of molecular and ecological data are increasingly accessible in microbial experimental ecosystems, it is worthwhile to tackle the questions of their complex hierarchical dynamics. Here, we propose an approach that combines microbial experimental ecosystems and a hierarchical dynamic model named population–reaction model. We present a simple microbial experimental ecosystem as an example and show how the system can be analyzed by a population–reaction model. We also show that population–reaction models can be applied to various ecological concepts, such as predator–prey interactions, climate change, evolution, and stability of diversity. Our approach will reveal a path to the general understanding of various ecosystems and organisms

SHc^c Realization of Minimal Model CFT: Triality, Poset and Burge Condition

Recently an orthogonal basis of $\mathcal{W}_N$-algebra (AFLT basis) labeled by $N$-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH$^c$ which is universal for any $N$. We show that it has an $\mathfrak{S}_3$ automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by $N$ or $M$ Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH$^c$. Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the $N$-Burge condition in the Hilbert space.Comment: 1+38 pages and 12 figures. v2: typos corrected + comments adde