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Electric analogy for Biot’s acousto-poroelasticity: application to equations of longitudinal poroelasto-electric waves in long bones
Published in J.-L. Auriault, C. Geindreau, P. Royer, J.-F. Bloch, C. Boutin, J. Lewandowska, Poromechanics II, 2020
The definition of the system of electromechanical analogies can be correctly introduced on the basis of the algebraic theory of dimensional analysis. The concept of dimensional space Π over the reals was introduced by Drobot (1954). The dimensions of all physical quantities are the elements of space Π. It is proved by Drobot that each dimensional space Π is isomorphic with a certain vector space. For this reason the dimensions can be considered as vectors. The other concepts concerning space Π such as the system of fundamental dimensions, the dimensional independence of elements and the dimensional combination of the dimensions correspond respectively to the following concepts from the vector space theory, namely: the basis of the vector space, the linear independence of vectors and the linear combination of vectors.
Mathematical Fundamentals to Inverse Problems
Published in Blaunstein Nathan, Yakubov Vladimir, Electromagnetic and Acoustic Wave Tomography, 2018
Vladimir Yakubov, Sergey Shipilov, Nathan Blaunstein
where U and V are unitary matrices of dimensions of (M × M) and (N × N), respectively. This means that UH · U = I and VH · V = I, where I is the identity matrix introduced in Section 1.3. The columns of matrices U and V are called, in algebraic theory, the and right singular vectors, respectively. Matrix S = [siδi,j] is a quasi-diagonal matrix, consisting of the singular values si, which are strictly positive and can usually be obtained one-by-one in decreasing of each value manner, and δi,j are the Kronecker symbols.
The history of Tutte–Whitney polynomials
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Tutte's PhD thesis was titled An Algebraic Theory of Graphs [T48]. The selection of topics is not typical of the field called “algebraic graph theory” today. Nonetheless, the title is apt, reflecting Tutte's aim of bringing much of graph theory into the kind of linear algebraic framework known today under the name of “representable matroids”. He develops this theory for over four hundred pages, culminating in his famous excluded minor characterizations of binary, graphic, and cographic matroids. Only Chapter V (the shortest in the thesis), “Chromatic functions”, concerns us here.
The Moran Spectrum as a Geoinformatic Tupu: implications for the First Law of Geography
Published in Annals of GIS, 2022
The SA Tupu presented in this paper indexes the complete sequence of Moran coefficients (MCs), the most widely used SA statistic, as a combination of diagrammatic, graphic, and numeric representations of SA structure for a given geographic landscape. It is therefore referred to as the Moran Spectrum. The bulk of the concepts and methods is based on an array of work involving the distribution of the MC (Tiefelsdorf and Boots 1995; de Jong, Sprenger, and Van Veen 1984) as well as the theory of Moran Eigenvector Spatial Filtering (MESF) (Griffith 1996, 2003, 2019) and Moran Eigenvector Maps (MEMs) in numeric ecology (Borcard and Legendre 2002). MESF and MEMs are rooted in an algebraic theory called eigen-decomposition that is viewed as being relatively complex by the geography community. Our goal is to facilitate the dissemination of these relatively complex concepts and methods by presenting them in the context of Geoinformatic Tupu, and in turn widening and deepening the applications of the FLG in modelling geospatial processes.
An integer linear programing approach to find trend-robust run orders of experimental designs
Published in Journal of Quality Technology, 2019
The literature on trend-robust run orders can be classified into three groups, depending on the solution method used. The first group consists of articles that use algebraic theory and/or foldover techniques. Cox (1951) was the first to study systematic run orders for an experiment with one qualitative two-level factor. Box and Hay (1953) study a biological experiment with one quantitative factor, a quadratic regression model in that factor, and a time trend that is approximated by a cubic polynomial. They use orthogonal polynomials to code the time trend. Daniel and Wilcoxon (1966) present some results for two-level full factorial and fractional factorial experiments. Their results are extended by Cheng and Jacroux (1988), who considered two-level factorial experiments with interactions. John (1990) constructs two-level and three-level factorial designs that are robust against linear and quadratic time trends. He considers main effects and interaction effects and uses a fold-over technique to find trend-robust run orders. Bailey, Cheng, and Kipnis (1992) present trend-robust run orders for two-level, multi-level, and mixed-level factorial designs. They use a generalized fold-over method.
Optimal control of non-minimum phase integrating processes with time delay using disturbance observer-based control scheme
Published in International Journal of Systems Science, 2018
Wei Zhang, Yagang Wang, Zhong Yin, Yongxiong Wang, Weidong Zhang
In this paper, a systematical DOB-control scheme for NMP integrating processes with time delay is proposed based on the algebraic theory. Both the set-point tracking specification and ILDR specification can be adjusted easily by simply tuning the performance degrees in the designed controller and filter of the DOB. The main contribution of this note is that the optimal filter of the DOB can be designed systematically by optimising the ILDR criterion. The designed controller and filter of the DOB have the feature of analytical and optimal. The design method is a specified weight design method. Specified weight functions are chosen for step inputs and inputs similar to steps. The design procedure is simple. The proposed DOB-control scheme can eliminate the influence of input load disturbance effectively and the quantitative performance specifications can be achieved easily by appropriately tuning the performance degrees and . To cope with the uncertainties caused by model uncertainties, parameters perturbation and approximation errors, the parameters tuning suggestion has been proposed to make the system achieve the desired dynamics of both set-point tracking and load disturbance rejection without loss of robustness requirements.