Tensor Decomposition With Generalized Lasso Penalties

We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multiway data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our approach presents many nontrivial challenges at the intersection of modeling and computation, which are studied in detail. An efficient coordinate-wise optimization algorithm for PTD is presented, and its convergence properties are characterized. The method is applied both to simulated data and real data on flu hospitalizations in Texas and motion-capture data from video cameras. These results show that our penalized tensor decomposition can offer major improvements on existing methods for analyzing multiway data that exhibit smooth spatial or temporal features.     

Atomic charge distribution in 4-isopropylphenol molecule derived from atomic polar tensors

On the basis of the calculated atomic polar tensors the generalized atomic polar tensor charges have been calculated for 4-isopropylphenol (4-IP) and related compounds: benzene, quinone, phenol and p-nitroaniline (p-NA). The second order Möller–Plesset perturbation method and Huzinaga–Dunning's double valence ζ basis set supplemented by d polarisation function on heavy atoms and p on hydrogen atoms (D95V^**) have been used. Analysis of the atomic charges has been done. It is found that the phenyl rings of the 4-IP and p-NA molecules have an intermediate structure between the aromatic ring and the quinoid one.     

Near-field optical microscopy of nonlinear susceptibilities

Local probing of second-order susceptibilities with a near-field optical microscope is demonstrated for the first time. Using an uncoated fiber tip as a light source, near-field images of a surface of y-cut LiNbO₃ crystal and of a multilayer Langmuir–Blodgett film of 2-docosylamino-5-nitropyridine are obtained at the fundamental pump and second harmonic wavelengths while simultaneously recording surface topography. It is shown that optical second-harmonic images for different polarizations of the pump light reflect the difference in magnitude of the corresponding components of the second-order susceptibility tensor. Various degrees of correlation in contrast of optical (fundamental and second harmonic) and topographical images are observed and discussed.     

Young symmetrizers and additive theory

For λ partition of m and A finite nonempty subset of a field we define the set of λ-restricted sums of m-tuples of elements of A, ∧ ^m λA, and using Additive Number Theory results from [J.A. Dias da Silva and Y.O. Hamidoune (<citeref rid="bib7">1990</citeref>). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (<citeref rid="bib8">1994</citeref>). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain a lower bound for its cardinality. Next, using results and techniques from [J.A. Dias da Silva and Y.O. Hamidoune (<citeref rid="bib7">1990</citeref>). A note on the minimal polynomial of the Kronecker sum of two linear operators. Linear Algebra Appl., 141, 283-287; J.A. Dias da Silva and Y.O. Hamidoune (<citeref rid="bib8">1994</citeref>). Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc., 26, 140-146] we obtain lower bounds for the degrees of minimal polynomials of restrictions of derivations to ranges of Young symmetrizers and to the symmetry class of tensors V _λ, and we show that the lower bound for the cardinality of ∧ ^m λA can also be obtained from these lower bounds.     

On multivariate polynomials in Bernstein-Bézier form and tensor algebra

The Bernstein-Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein-Bézier form. As an application we consider Hermite interpolation with polynomials and splines.     

Prehomogeneous tensor spaces

We refine the classification of prehomogeneous vector spaces provided by Sato and Kimura in the case of tensor spaces, presenting a quick way to determine whether a given tensor space is prehomogeneous or not.     

Diagrams,Tensors and Geometric Reasoning

Geometry and in particular projective geometry (and its corresponding invariant theory) deals a lot with structural properties of geometric objects and their interrelations. This papers describes how concepts of tensor calculus can be used to express geometric invariants and how, in particular, diagrammatic notation can be used to deal with invariants in a highly intuitive way. In particular we explain how geometries like euclidean or spherical geometry can be dealt with in this framework. Dedicated to the memory of Victor Klee, and in particular to his striving for conceptual simplicity     

Polynomial integration over the unit circle

In this work, recursive formulas facilitating the computation of very complex tensor integrals over the unit circle, are obtained by making use of the divergence theorem and mathematical induction. Using these formulas, the integrals over the unit circle of polynomials with any high degree are easily computed.