We consider the uniqueness of solution (nonsingularity) of systems of r generalized Sylvester and ⋆-Sylvester equations with n × n coefficient matrices. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix penc...

We study systems of three bivariate polynomials whose Newton polygons are scaled copies of a single polygon. Our main contribution is to construct square resultant matrices, which are submatrices of those introduced by Cattani et al. (1998), and whose determinants are nontrivial multiples of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type ...

We present parallel algorithms for triangular periodic Sylvester-type matrix equations, conceptually being the third step of a periodic Bartels–Stewart-like solution method for general periodic Sylvester-type matrix equations based on variants of the periodic Schur decomposition. The presented algorithms are designed and implemented in the framework of the recently developed HPC library SCASY a...

The first step in the generalization of the classical theory of homogeneous equations to the case of arbitrary support is to consider algebraic systems with multihomogeneous structure. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, and including matrices of hybrid type of these two. Our appro...

We continue our presentation of parallel ScaLAPACK-style algorithms for solving Sylvester-type matrix equations. In Part II, we present SCASY, a state-of-the-art HPC software library for solving 44 sign and transpose variants of eight common standard and generalized Sylvester-type matrix equations. The internal design of the library, Fortran interfaces and implementation issues are discussed in...

We introduce Gaussian Process Topic Models (GPTMs), a new family of topic models which can leverage a kernel among documents while extracting correlated topics. GPTMs can be considered a systematic generalization of the Correlated Topic Models (CTMs) using ideas from Gaussian Process (GP) based embedding. Since GPTMs work with both a topic covariance matrix and a document kernel matrix, learnin...

Ion mobility spectrometry (IMS) has been widely used for the on-site detection of trace chemicals, but continue to suffer from a low duty cycle ion injection. The Hadamard transform (HT-IMS) technique was employed address problem with increased signal-to-noise ratio (SNR). However, in this work, through simulation, certain deviation between mathematical principle and actual data collection proc...

This paper addresses two problems: an image denoising problem assuming dense observations and an image reconstruction problem from sparse data. It shows that both problems can be solved by the Sylvester/Lyapunov algebraic equation. The Sylvester/Lyapunov equation has been extensively studied in Control Theory and it can be efficiently solved by well known numeric algorithms. This paper proposes...