Co-clustering of Time-Dependent Data via the Shape Invariant Model

Multivariate time-dependent data, where multiple features are observed over time for a set of individuals, are increasingly widespread in many application domains. To model these data, we need to account for relations among both time instants and variables and, at the same time, for subject heterogeneity. We propose a new co-clustering methodology for grouping individuals and variables simultaneously, designed to handle both functional and longitudinal data. Our approach borrows some concepts from the curve registration framework by embedding the shape invariant model in the latent block model, estimated via a suitable modification of the SEM-Gibbs algorithm. The resulting procedure allows for several user-defined specifications of the notion of cluster that can be chosen on substantive grounds and provides parsimonious summaries of complex time-dependent data by partitioning data matrices into homogeneous blocks. Along with the explicit modelling of time evolution, these aspects allow for an easy interpretation of the clusters, from which also low-dimensional settings may benefit.

Variant-Coherence Gaussian Sources

The celebrated Gaussian Schell model source with its shift-invariant degree of coherence may be the basis for devising sources with space-variant properties in the spirit of structured coherence. Starting from superpositions of Gaussian Schell model sources, we present two classes of genuine cross-spectral densities whose degree of coherence varies across the source area. The first class is based on the use of the Laplace transform while the second deals with cross-spectral densities that are shape-invariant upon paraxial propagation. For the latter, we present a set of shape-invariant cross-spectral densities for which the modal expansion can be explicitly found. We finally solve the problem of ascertain whether an assigned cross-spectral density is shape-invariant by checking if it satisfies a simple differential constraint.

Anti-PT Transformations and Complex PT-Symmetric Superpartners

A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.

A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams

A new class of sources presenting structured coherence properties is introduced and analyzed. They are obtained as the incoherent superposition of coherent Laguerre-Gaussian modes with suitable coefficients. This ensures that the shape of the intensity profile and the spatial coherence features of the propagated beams are invariant during paraxial approximation. A simple analytical expression is obtained for the cross-spectral density of the sources of this class, regardless of the number of superposed modes. Properties of these sources are analyzed and described by several examples.