Standard Matrix Representation Research Articles

Kinematics Analysis and Trajectory Planning Computer Simulation of Smart Apple Bagging Robot by Hail Suppression

In view of the high labor intensity and low efficiency of artificial anti hail apple bagging work, it needs a lot of manpower and financial resources. Therefore, Gansu Zede Electronic Technology Co., Ltd. research and development suitable for the actual production of anti hail apple bagging robot has broad application prospects. In this paper, the kinematics model is established based on the standard matrix representation, and the forward kinematics and inverse kinematics of the anti hail apple bagging robot manipulator are solved and analyzed, and the robot toolbox based on MATLAB is used for simulation verification and joint space trajectory planning simulation.

Symmetries of the Eikonal equation

The infinitesimal algebra of Lie symmetries of the Eikonal equation is shown to be isomorphic to o(n+1,2) when there are n independent variables. An explicit basis is found that is aligned with the standard basis coming from the standard matrix representation of o(n+1,2) thereby making it possible to read off inequivalent one-dimensional symmetry vector fields. The symmetries are used to construct various solutions of the Eikonal equation.

A Splitter Theorem Relative to a Fixed Basis

A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M)–B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M, it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N-minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s 1 and s 2, such that for each i ∈ {1, 2}, si(M/s i ) is 3-connected and has an N-minor when s i ∈ B, and co(M\s i ) is 3-connected and has an N-minor when s i ∈ E(M)–B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N-minor, then (P, E(M)–P) is a sequential 3-separation.

Coined quantum walks lift the cospectrality of graphs and trees

In this paper we explore how a spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations. The algorithm runs in polynomial time and, moreover, can distinguish many graphs for which there is no subexponential time algorithm that is proven to be able to distinguish between them. In the paper, we give a description of the coined quantum walk from the field of quantum computing. The evolution of the walk is governed by a unitary matrix. We show how the spectrum of this matrix is related to the spectrum of the transition matrix of the classical random walk. However, despite this relationship the behaviour of the quantum walk is vastly different from the classical walk. This leads us to define a new matrix based on the amplitudes of paths of the walk whose spectrum we use to characterise graphs. We carry out three sets of experiments using this matrix representation. Firstly, we test the ability of the spectrum to distinguish between sets of graphs that are cospectral with respect to standard matrix representation. These include strongly regular graphs, and incidence graphs of balanced incomplete block designs (BIBDs). Secondly, we test our method on ALL regular graphs on up to 14 vertices and ALL trees on up to 24 vertices. This demonstrates that the problem of cospectrality is often encountered with conventional algorithms and tests the ability of our method to resolve this problem. Thirdly, we use distances obtained from the spectra of S + ( U 3 ) to cluster graphs derived from real-world image data and these are qualitatively better than those obtained with the spectra of the adjacency matrix. Thus, we provide a spectral representation of graphs that can be used in place of standard spectral representations, far less prone to the problems of cospectrality.

Maintaining 3-connectivity relative to a fixed basis

A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of E ( M ) − B . If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M.

SDM: A Fast Distance-Based Approach for (Super)Tree Building in Phylogenomics

Phylogenomic studies aim to build phylogenies from large sets of homologous genes. Such "genome-sized" data require fast methods, because of the typically large numbers of taxa examined. In this framework, distance-based methods are useful for exploratory studies and building a starting tree to be refined by a more powerful maximum likelihood (ML) approach. However, estimating evolutionary distances directly from concatenated genes gives poor topological signal as genes evolve at different rates. We propose a novel method, named super distance matrix (SDM), which follows the same line as average consensus supertree (ACS; Lapointe and Cucumel, 1997) and combines the evolutionary distances obtained from each gene into a single distance supermatrix to be analyzed using a standard distance-based algorithm. SDM deforms the source matrices, without modifying their topological message, to bring them as close as possible to each other; these deformed matrices are then averaged to obtain the distance supermatrix. We show that this problem is equivalent to the minimization of a least-squares criterion subject to linear constraints. This problem has a unique solution which is obtained by resolving a linear system. As this system is sparse, its practical resolution requires O(naka) time, where n is the number of taxa, k the number of matrices, and a < 2, which allows the distance supermatrix to be quickly obtained. Several uses of SDM are proposed, from fast exploratory studies to more accurate approaches requiring heavier computing time. Using simulations, we show that SDM is a relevant alternative to the standard matrix representation with parsimony (MRP) method, notably when the taxa sets of the different genes have low overlap. We also show that SDM can be used to build an excellent starting tree for an ML approach, which both reduces the computing time and increases the topogical accuracy. We use SDM to analyze the data set of Gatesy et al. (2002, Syst. Biol. 51: 652-664) that involves 48 genes of 75 placental mammals. The results indicate that these genes have strong rate heterogeneity and confirm the simulation conclusions.

The Information Content of Trees and Their Matrix Representations

Any tree can be represented by a variety of matricesfrom which the tree can be reconstructed. Such matrixrepresentations (encodings) of trees have been used in anumber of contexts, including measuring tree similarity(Farris, 1973), cophylogeny (Brooks, 1981), and consen-sus (Nelson and Ladiges, 1994b; Phillips and Warnow,1996),buttheyhavebecomemoreprominentthroughthematrix representation with parsimony (MRP) approachto supertree construction (Baum, 1992; Ragan, 1992;Sanderson et al., 1998). Most published supertrees haveutilized MRP (e.g., Jones et al., 2002; Kennedy and Page,2002; Pisani et al., 2002; Purvis, 1995a), in which matrixrepresentations of trees are combined into a compositematrix and analyzed with parsimony. In standard MRP,the elements of matrix representations are the relation-ships given by the full splits or bipartitions of leaves (ter-minal taxa) present in the trees. In the context of rootedtrees, standard matrix representations have one binary(pseudo)characterforeachcladeinwhichclademembersarescoredconventionally(andarbitrarily)as‘1’andnon-members and the root (MRP outgroup) as ‘0’ (Table 1).Purvis (1995b) developed an alternative matrix rep-resentation (Purvis coding) intended to compensate forapparent overweighting of larger trees in standard MRP.His method, which is applicable only to rooted trees, at-tempted to remove redundant information in standardmatrix representations, but was criticized by Ronquist(1996)whoarguedthattheinformationremovedwasnotredundant. Williams and Humphries (2003) advocatedthe use of matrix representations based on three-itemstatements or triplets (Nelson and Ladiges, 1992, 1994b;Nelson and Platnick, 1991; Wilkinson et al., 2001), whichare also applicable only to rooted trees, and they usedsuch matrices, and an associated differential weighting,as a yardstick to compare and contrast standard andPurvis matrix representations. Matrix representationscan also be based on pairwise pathlength distances be-tween leaves (Lapointe and Cucumel, 1997; Lapointe etal., 2003) and nonbinary discrete characters (Semple andSteel, 2002).We believe the literature on the relative merits of al-ternative binary matrix representations of trees to beconfused and confusing. Here we use a quantitativemeasure, cladistic information content (CIC; Thorleyet al., 1998), to compare the information content of treesand their matrix representations. Specifically, we revisitRonquist’s (1996) critique of Purvis coding. We demon-strate that the information removed by this method isredundant in the sense that Purvis (1995b) intended, butthat it does not remove all such redundant informationasPurvissupposed.BuildingonRonquist(1996),wedis-cuss the distinction between the representation of a treeand the representation of the data from which a tree hasbeen inferred and its importance in choosing a matrixrepresentation. This leads us to consider logical relationsand dependencies among pseudocharacters from thesametree,andhighlightfundamentalinferencerulesthatform the basis of the logical calculus of cladistic relation-ships. We show that fractional weighting as proposedby Nelson and Ladiges (1992), and the form of this usedby Williams and Humphries (2003), does not account forall logical dependencies between triplets. This leads usto propose an alternative weighting scheme. Finally, wereview the results and present a critical discussion ofWilliams and Humphries’ (2003) comparison of alterna-tive methods. We use the terms triplet and three-item

Modelling and simulation of particle breakage in impact crushers

In this paper, we present a phenomenological model that predicts the size distribution of the product issued from impact crushing in function of the rotor strike radius and velocity, the material properties and size distribution of the feed as well as the feed rate. The model is based on the standard matrix representation including classification and breakage matrices. It can be applied to both horizontal- and vertical-shaft impact crushers by means of the corresponding estimations for the average impact energy per unit mass presented here. We propose a new classification function for impact crushers in the form of a Weibull cumulative distribution. The minimum size of the particles that undergo breakage is assumed to be a function of the impact energy and the feed rate. The model predictions are compared with experimental data obtained for limestone treated in a pilot-plant hammer crusher. The dependence of the product size distribution on the rotor velocity is investigated. The influence of the feed rate on the product size is also simulated.

Matrix representations and parameterizations of lossless and reciprocal multiports

Stimulated by recent work on the synthesis of a certain class of surface acoustic wave (SAW) filters, we discuss the exceedingly classical subject of physical realizability criteria for lossless, and reciprocal linear multiports. We give an essentially coordinate-free formulation of the realizability conditions for passive, lossless and reciprocal multiports. For this sake we describe a linear multiport by its external behavior (the linear subspace generated by all admissible signal pairs at the ports) and analyze the properties of this space in terms of Gramians of its bases with respect to physically relevant metrics. The standard matrix representations (impedance matrix, scattering matrix, . . . ) arise by the introduction of various affine coordinates in the external behavior. Instead of discussing their well-known coordinate-dependent properties we focus on the so called P -matrix as an attractive non-standard example and study its properties and relations to other matrix representations. As a major result we show that the parameterization of lossless multiports as induced by the P -matrix is largely equivalent to the Arov-Dewilde-Dym parameterization of J-inner functions.

Trajectories of fertility and household composition in the demographic profile of the Czech Republic.

The household composition matrix is a representation of the demographic structure of households, specific to age groups of household members and household heads. As such, the matrix reflects also the environmental conditions, housing in particular, that mould households' demographic structure. By specifically depicting the presence of children in households, household composition could be viewed as gauging fertility within the context of housing conditions. This stance is examined in an application to Czech census data for the year 1991, at the commencement of an intense process of socio-economic transformation that accompanied the collapse of communism across Eastern Europe. Within this process, housing had an inadvertent impact upon the structure of households in general, and upon fertility decline in particular. By using the standard matrix representation of household composition, correspondence between trajectories of age-specific fertility and household composition emerge throughout the Czech Republic. This correspondence illustrates the potential household composition analysis carries for fertility measurement and estimation in rapidly changing economic environments.

Stable Norms on Complex Numbers and Quaternions

In this paper, we study stability properties of norms on the complex numbers and on the quaternions. Our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable. Part of the paper is devoted to the standard matrix representations of the above number systems, where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant. We conclude by considering proper seminorms, observing that none are stable on the complex numbers or on the quaternions.

Controllability on classical Lie groups

We consider the problem of accessibility and controllability of certain bilinear systems. These evolve on Lie groups whose Lie algebras are the normal real forms of complex simple Lie algebras. Previous results by other authors were obtained under the assumption that the controlled vector field is strongly regular. Our paper is aimed at weakening this requirement, and involves relating the root structure of elements in a Lie algebra as above to the nodal connection graphs obtained from their standard matrix representations. This is in turn related to a standard irreducibility assumption on the uncontrolled vector field. The abstract results on generation of Lie algebras are of some independent interest.