Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

Implicitization of curves and (hyper)surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive

Localization and Pattern Formation in Quantum Physics. II. Waveletons in Quantum Ensembles

In this second part we present a set of methods, analytical and numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. The key points demonstrating advantages of this approach are: (i) effects of localization of possible quantum states; (ii) effects of non-perturbative multiscales which cannot be calculated by means of perturbation approaches; (iii) effects of formation of complex/collective quantum patterns from localized modes and classification and possible control of the full zoo of quantum states, including (meta) stable localized patterns (waveletons). We demonstrate the appearance of nontrivial localized (meta) stable states/patterns in a number of collective models covered by the (quantum)/(master) hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of ``wignerization'' procedure (Weyl-Wigner-Moyal quantization) of classical BBGKY kinetic hierarchy, and present the explicit constructions for exact analytical/numerical computations (fast convergent variational-wavelet representation). Numerical modeling shows the creation of different internal structures from localized modes, which are related to the localized (meta) stable patterns (waveletons), entangled ensembles (with subsequent decoherence) and/or chaotic-like type of behaviour.Comment: LaTeX2e, spie.cls, 13 pages, 6 figures, submitted to Proc. of SPIE Meeting, The Nature of Light: What is a Photon? Optics & Photonics, SP200, San Diego, CA, July-August, 200