To every k-dimensional modular invariant vector space we associate a modular form on SL(2,Z) of weight 2k. We explore number theoretic properties of this form and found a sufficient condition for its vanishing which yields modular identities (e.g., Ramanujan-Watson’s modular identities). Furthermore, we focus on a family of modular invariant spaces coming from suitable two-dimensional spaces vi...

The issue of how to deal with the modular transformations – large diffeomorphisms – in (2+1)-quantum gravity on the torus is discussed. I study the Chern-Simons/connection representation and show that the behavior of the modular transformations on the reduced configuration space is so bad that it is possible to rule out all finite dimensional unitary representations of the modular group on the ...

Interconnect congestion estimation plays an important role in design automation of VLSI designs. This paper presents a novel probabilistic approach to predict the wiring space in two-dimensional arrays. We propose a hierarchical estimation method to derive approximated upper bounds for the wiring space, and we use the net density distribution to predict the routing congestion. Experimental resu...

State-space models (SSMs) are a highly expressive model class for learning patterns in time series data and for system identification. Deterministic versions of SSMs (e.g., LSTMs) proved extremely successful in modeling complex timeseries data. Fully probabilistic SSMs, however, unfortunately often prove hard to train, even for smaller problems. To overcome this limitation, we propose a scalabl...

This paper is focused on the kinematic control of redundant modular robots for trajectory tracing. Based on the geometric numerical inverse kinematic algorithm developed for modular robots, a new online control method is presented. In this method, the inverse kinematic solution can be optimized through constructing a weighted matrix. Following this approach, some fundamental interpolation algor...

In this paper we present a fixed point theorem of Banach type in modular space. We give an application of this result to a nonlinear integral equation in Musielak-Orlicz space. 0. Introduction It is well known that one of the standard proofs of Banach’s fixed point theorem is based on Cantor’s theorem in complete metric spaces [3, 4]. To this end, using some convenient constants in the contract...