Graphs and Matrices

The adjacency matrix operations,which connect with configuration transformation correspondingly,can be used for analysis of configuration transformation of metamorphic mechanisms and the corresponding algorithm can easily be simulated by computer.But the adjacency matrix based on monochrome topological graph is not suitable for the topological representation of mechanisms with multiple joints.The method of adjacency matrix operations has its own limitations for analysis of configuration transformation of metamorphic mechanisms because it can only be used in the topological representation of mechanisms with single joints.In order to overcome the drawback of the adjacency matrix,a kind of new matrix named as extended adjacency matrix is proposed to express topological structures of all mechanisms.The extended adjacency matrix is not only suitable for the topological representation of mechanisms with single joints,but also can be used in that of mechanisms with multiple joints.On this basis,a method of matrix operations based on the extended adjacency matrix is proposed to analyze the configuration transformation of metamorphic mechanisms.The method is not only suitable for configuration analysis of metamorphic mechanisms with single joints as well as metamorphic mechanisms with multiple joints.The method is evaluated by calculating two examples representing metamorphic mechanisms with single joint and multiple joints respectively.It can be concluded that the method is effective and correct for analysis of configuration transformation of all metamorphic mechanisms.The proposed method is simple and easy to be achieved by computer programming.It provides a basis for structural synthesis of all metamorphic mechanisms.

We propose a fundamentally new approach to Datalog evaluation. Given a linear Datalog program DB written usingNconstants and binary predicates, we first translate if-and-only-if completions of clauses in DB into a setEq(DB) of matrix equations with a non-linear operation, where relations inMDB, the least Herbrand model of DB, are encoded as adjacency matrices. We then translateEq(DB) into another, but purely linear matrix equationsẼq(DB). It is proved that the least solution ofẼq(DB) in the sense of matrix ordering is converted to the least solution ofEq(DB) and the latter givesMDBas a set of adjacency matrices. Hence, computing the least solution ofẼq(DB) is equivalent to computingMDBspecified by DB. For a class of tail recursive programs and for some other types of programs, our approach achievesO(N3) time complexity irrespective of the number of variables in a clause since only matrix operations costingO(N3) or less are used. We conducted two experiments that compute the least Herbrand models of linear Datalog programs. The first experiment computes transitive closure of artificial data and real network data taken from the Koblenz Network Collection. The second one compared the proposed approach with the state-of-the-art symbolic systems including two Prolog systems and two ASP systems, in terms of computation time for a transitive closure program and the same generation program. In the experiment, it is observed that our linear algebraic approach runs 101~ 104times faster than the symbolic systems when data is not sparse. Our approach is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

Bond‐extended stochastic and nonstochastic bilinear indices are introduced in this article as novel bond‐level molecular descriptors (MDs). These novel totals (whole‐molecule) MDs are based on bilinear maps (forms) similar to use defined in linear algebra. The proposed nonstochastic indices try to match molecular structure provided by the molecular topology by using the k th Edge(Bond)‐Adjacency Matrix ( E k , designed here as a nonstochastic E matrix). The stochastic parameters are computed by using the k th stochastic edge‐adjacency matrix, ES k , as matrix operators of bilinear transformations. This new edge (bond)‐adjacency relationship can be obtained directly from E k and can be considered like a new matrix‐transformation strategic to obtain new relations for a molecular graph. In both set of MDs, chemical information is codified by using different pair combinations of atomic weightings (in this case four atomic‐labels: atomic mass, polarizability, van der Waals volume, and electronegativity). In addition, a local‐fragment (bond‐type) formalism was also developed. The k th bond‐type bilinear indices are calculated by summing the k th bond bilinear indices of all bonds of the same bond type in the molecules. The new set of MDs can be easily and quickly calculated in our in house software TOMOCOMD‐CARDD (topological molecular computational design computer‐aided‐rational‐drug design). The reported application and utilization of these MDs for predictive capability correlations of structure with physicochemical and pharmacological properties are reviewed. Three benchmark datasets have been used to evaluate the QSPR/QSAR behavior of the new bond‐level TOMOCOMD‐CARDD MDs. We developed the QSPR models to describe several physicochemical properties of octane isomers (First Case) and, to analyze of the boiling point of 28 alkyl‐alcohols (Second Case) and to examine of the specific rate constant (log k ), the partition coefficient (log P ), as well as the antibacterial activity of 34 derivatives of 2‐furylethylenes (Third Case). For these three rounds, the quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. A leave‐one‐out cross‐validation procedure revealed that the regression models had a good predictability. The comparison with other approaches reveals good performance of the method proposed. Therefore, it is clearly demonstrated that this suitability is higher than that shown by other 2D/3D well‐known sets of MDs. The approach described here appears to be a very promising structural invariant, useful for QSPR/QSAR studies and shown to provide an excellent alternative or guides for discovery and optimization of new lead compounds, reducing the time and cost of the traditional procedure. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

In the realm of image classification, the capsule network is a network topology that packs the extracted features into many capsules, performs sophisticated capsule screening using a dynamic routing mechanism, and finally recognizes that each capsule corresponds to a category feature. Compared with previous network topologies, the capsule network has more sophisticated operations, uses a large number of parameter matrices and vectors to express picture attributes, and has more powerful image classification capabilities. However, in the practical application field, the capsule network has always been constrained by the quantity of calculation produced by the complicated structure. In the face of basic datasets, it is prone to over-fitting and poor generalization and often cannot satisfy the high computational overhead when facing complex datasets. Based on the aforesaid problems, this research proposes a novel enhanced capsule network topology. The upgraded network boosts the feature extraction ability of the network by incorporating a multi-scale feature extraction module based on proprietary star structure convolution into the standard capsule network. At the same time, additional structural portions of the capsule network are changed, and a variety of optimization approaches such as dense connection, attention mechanism, and low-rank matrix operation are combined. Image classification studies are carried out on different datasets, and the novel structure suggested in this paper has good classification performance on CIFAR-10, CIFAR-100, and CUB datasets. At the same time, we also achieved 98.21% and 95.38% classification accuracy on two complicated datasets of skin cancer ISIC derived and Forged Face EXP.

For the problem of computing the rank of a matrix we have a complexity result and a practical implementation, both of which apply best to the case of a matrix whose rank is substantially smaller than its order.First, matrix rank can be computed in essentially optimal time when the rank is sufficiently small, specifically when the rank is less than the matrix order to the two-thirds power.Second, we have built software infrastructure to improve efficiency of matrix operations over a small finite field. Using this we have computed the 3-ranks of the first 7 matrices of the sequence D(3,k). These are adjacency matrices of strongly regular graphs defined by a construction of Dickson based on operations in a semifield of order 32k.In particular, D(3,7) is a 4,782,969 by 4,782,969 matrix and it's rank computation uses four days of CPU time.On the basis of the first 7 values, we conjecture that the sequence of ranks of the D(3,k) satisfies a linear recurrence of degree three.

This paper presents adjacency matrix originated three-dimensional matrix for structural representation of mechanisms by integrating binary string in the sense of displacement subgroup theory. The improved elementary matrix operation for expressing of topological transformation of metamorphic mechanism is integrated in the three-dimensional matrix representation. A novel reconfigurable eight-bar linkage employing variable-axis revolute joints is proposed and the three-dimensional matrix for each working phase is expressed to reveal the distinct geometry. The improved elementary matrix operation is used to express the topological transformation of a metamorphic eight-bar linkage and to identify the effectiveness of the operation.