Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

Homotopy techniques for tensor decomposition and perfect identifiability

LetTbe a general complex tensor of format{(n_{1},\dots,n_{d})}. When the fraction{\prod_{i}n_{i}/[1+\sum_{i}(n_{i}-1)]}is an integer, and a natural inequality (called balancedness) is satisfied, it is expected thatThas finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions ofT, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats{(3,4,5)}and{(2,2,2,3)}which have a unique decomposition as the sum of six and four decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the onlyidentifiablecases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.

Skew-symmetric tensor decomposition

We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in [Formula: see text] with [Formula: see text] and [Formula: see text]. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.

Methods and Algorithms of Speech Signals Processing and Compression and Their Implementation in Computer Systems

The review and comparative analysis of the methods of compression and recognition of speech signals is carried out. The result of the carried out analysis of the existing recognition methods indicates, that all of them are based on the use of “inflexible” algorithms, which are badly adapted to the characteristic features of speech signals, thus degrading the efficiency of the operation of the whole recognition system. The necessity of the use of algorithms for determination of recognition features along with the use of the wavelet packet analysis as one of the advanced directions of the creation of the effective methods and principles of the development of the speech signals recognition systems is substantiated. Analysis of the compression methods with the use of the orthogonal transformations at the complete exception of minimal decomposition factors is conducted; a maximal possible compression degree is defined. In this compression method the orthogonal transformation of the signal segment with the subsequent exception of the set of the smallest modulo decomposition factors, irrespective of the order of their distribution, is conducted. Therefore the additional transfer of the information on the factors distribution is required. As a result, two information streams appear, the first one corresponds to the information stream on the decomposition factors, and the second stream transfers information on the distribution of these factors. Method of the determination of the speech signals recognition features and the algorithm for nonlinear time normalization is proposed and proved. Wavelet-packet transformation is adaptive, i.e. it allows adapting to the signal features more accurately by means of the choice of the proper tree of the optimal decomposition form, which provides the minimal number of wavelet factors at the prescribed accuracy of signal reconstruction, thus eliminating the information-surplus and unnecessary details of the signals. Estimation of the informativeness of the set of wavelet factors is accomplished by the entropy. In order to obtain the recognition factors, the spectral analysis operation is used. In order to carry out the temporary normalization, the deforming function is found, the use of which minimizes the discrepancy between the standard and new words realization. Dedicated to the determination of admissible compression factors on the basis of the orthogonal transformations use at the incomplete elimination of the set of minimal decomposition factors, to the creation of the block diagram of the method of the recognition features formation, to the practical testing of the software- methods. In order to elevate the compression factor, the adaptive uniform quantization is used, where the adaptation is conducted for all the decomposition factors. The program testing of the recognition methods is carried out by means of determination of the classification error probability using Mahalanobis (Gonzales) distance.

MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

Let C ⊂ ℙn+1 be a rational normal curve and let X ⊂ ℙn be one of its tangential projection. We describe the X-rank of a point P ∈ ℙn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.

Improvement of H2O2 Stabilization in Rainwater Samples

Peroxides collected in rainwater readily decompose in the natural matrix, consequently it is hard to estimate their values present during precipitation events. For this reason, only few data concerning the concentration of aqueous H2O2 in dew, cloud and rainwater are available. Therefore, it is necessary to establish a sampling protocol under conditions that assure the real-time estimation of hydrogen peroxide in rainwater samples. Hydrogen Peroxide was determined by a fluorometric method in rainwater collected in fives sites in Mexico. Measured levels for all sampled sites were in agreement with values reported by other authors for marine and continental sites. To establish a sampling protocol, several methods reported for preservation of H2O2 in rainwater were assessed. Sampling protocol proposed in this research involves forming fluorescent dimmer on-site immediately after sample collection, and storing at dark and 4 ºC. This procedure gives a 10 days margin to perform sample analysis, guarantying a minimal decomposition of H2O2 in rainwater samples (approximately 10%) and allowing more time to perform sampling campaigns even in remote sites. The other methods evaluated (freezing, filtering, using acidified samplers, adding sodium stannate, and storing samples and their fluorescent dimmer at 0ºC) do not exert any preservation effect over H2O2 levels in rainwater samples.