In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple αδ of the divisor δ of singular curves as the boundary divisor, construct the log canonical model for the pair (M3, αδ) using geometric invariant theory as we vary α from one to zero, and give a modular interpretation of each log canonical model...

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for cer...

This paper shows how the modular structure of composite systems can guide the state-space exploration in explicit-state linear-time model-checking and make it more efficient in practice. Given a composite system where every module has input and output variables — and variables of different modules can be connected — a total ordering according to which variables are generated is determined, thro...

The role of the modular group in the holonomy representation of (2+1)-dimensional quantum gravity is studied. This representation can be viewed as a “Heisenberg picture,” and for simple topologies, the transformation to the ADM “Schrödinger picture” may be found. For spacetimes with the spatial topology of a torus, this transformation and an explicit operator representation of the mapping class...

H(n) = uniform bound for the number of limit cycles of (1) . One way to formulate the Hilbert 16th problem is the following: Hilbert 16th Problem (HP). Estimate H(n) for any n ∈ Z+. To prove that H(1) = 0 is an exercise, but to find H(2) is already a difficult unsolved problem (see [DRR,DMR] for work in this direction). Below we discuss two of the most significant branches of research HP has ge...

H. Gardner introduced the idea of multiple intelligence. One type of these intelligences is the spatial intelligenc. P. H. Maier in his thesis distinguished five elements of the spatial intelligence: spatial perception, visualization, mental rotation, spatial relation, and spatial orientation. Maier showed a modular construction system of polyhedrons, and different activities to improve the stu...

Even in state-spaces of modest size, planning is plagued by the “curse of dimensionality”. This problem is particularly acute in human and animal cognition given the limited capacity of working memory, and the time pressures under which planning often occurs in the natural environment. Hierarchically organized modular representations have long been suggested to underlie the capacity of biologic...

The verification of dynamic properties of a reactive systems by model-checking leads to a potential combinatorial explosion of the state space that has to be checked. In order to deal with this problem, we define a strategy based on local verifications rather than on a global verification. The idea is to split the system into subsystems called modules, and to verify the properties on each modul...

This paper deals with the automatic dependability analysis of systems designed using UML. An automatic transformations is defined for the generation of models to capture systems dependability attributes, like reliability. The transformation concentrates on structural UML views, available early in the design, to operate at different levels of refinement, and tries to capture only the information...