In this paper, we define invariant tensor product and study invariant tensor products associated with discrete series representations. Let G(V1) × G(V2) be a pair of classical groups diagonally embedded in G(V1 ⊕ V2). Suppose that dimV1 < dimV2. Let π be a discrete series representation of G(V1 ⊕ V2). We prove that the functor π⊗G(V1) maps unitary representations of G(V1) to unitary representat...

Let G be a noncompact simple Lie group with finite center, let K be a maximal compact subgroup, and suppose that rank G = rank K. If G/K is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and it occurs exactly twice in the largest root. Le...

Learning useful representations without supervision remains a key challenge in machine learning. In this paper, we propose a simple yet powerful generative model that learns such discrete representations. Our model, the Vector QuantisedVariational AutoEncoder (VQ-VAE), differs from VAEs in two key ways: the encoder network outputs discrete, rather than continuous, codes; and the prior is learnt...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...

This study was conducted to evaluate the cultural contents and cultural elements represented in Vision (I & II) and Pre-University English (I & II). This study also aimed at determining if there were significant differences between these series in terms of cultural content and cultural element representation. To this end, the two ELT coursebook series were selected and analyzed based on Cortazz...

The paper presents realization theory of discrete-time linear switched systems (abbreviated by DTLSSs). We present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we pr...

This paper presents a subband approach to financial time series prediction. Multivariate empirical mode decomposition MEMD is employed here for multiband representation of multichannel financial time series together. Autoregressivemoving average ARMA model is used in prediction of individual subband of any time series data. Then all the predicted subband signals are summed up to obtain the over...

due to the changes which may occur in their parameters, systems are usually demonstrated by some subsystems for different conditions. this paper employs modified petri nets (mpn) to model theses subsystems and makes it simple to analyze them. in this method, first, the continuous transfer function is converted to a discrete transfer function and then, by mpn, system is modeled and analyzed. all...

The use of the Hilbert transform for time/frequency analysis of signals is briefly considered. While the Hilbert transform is based on arbitrary continuous signals, most practical signals are digitially sampled and time-limited. To avoid aliasing in the sampling process the signals must also be bandlimited. It is argued that it is reasonable to consider such sampled signals as periodic (this is...