Search

The matrix expression, topological index and atomic attribute of molecular topological structure are reviewed. Nine ma- trices, twenty-six kinds of indices and eight methods dealing with weighted molecular graphs are summed up in three tables. Some shortcomings of the topological indices are discussed as: (1) the physical-chemical meaning of topological index is not explicit; (2) it is difficult to interpret the QSAR and QSPR models derived from the topological indices; and (3) topological index usually neglects the stereochemical information or the three-dimensional structure of the molecule. Three directions of topological index are focused on: (1) description of local information; (2) studies on inter-correlation of topological index; and (3) variable index.

This paper investigates the optimization of traffic congestion systems via network congestion game approach. Firstly, using the semi-tensor product(STP) of matrices, the matrix expression of network congestion game is obtained. Secondly, a necessary and sufficient condition is proposed to guarantee that the traffic systems can be transformed into network congestion game with given performance criterion as its weighted potential function. Then an algorithm is provided to design the traffic congestion price in the case that conversion can be established. Thirdly, by designing proper learning rule, the optimization of traffic systems can be achieved when individuals optimize their own utility function. Moreover, two special cases which make our results more accord with reality and rich. Finally, an example is exploited to demonstrate the effectiveness of our obtained results.

In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained.

The kinematic analysis of the MPM uses the redundant parameters method that allow obtaining analytical description of links position in any mechanisms and manipulators, as well as obtaining more simple final matrix expression (with smaller number of terms). Using this method, the direct and inverse kinematics problem of the SHOLKOR parallel manipulator is solved.

This paper investigates the networked evolutionary model based on snow-drift game with the strategy of rewards and penalty. Firstly, by using the semi-tensor product of matrices approach, the mathematical model of the networked evolutionary game is built. Secondly, combined with the matrix expression of logic, the mathematical model is expressed as a dynamic logical system and next converted into its evolutionary dynamic algebraic form. Thirdly, the dynamic evolution process is analyzed and the final level of cooperation is discussed. Finally, the effects of the changes in the rewarding and penalty factors on the level of cooperation in the model are studied separately, and the conclusions are verified by examples.

The purpose of this paper is to study a new generalized inverse called weighted CMP inverse associated with an operator between two Hilbert spaces, using its Wg-Drazin inverse and its Moore–Penrose inverse. This generalized inverse extends the notion of the weighted CMP inverse for a rectangular matrix. We give some new properties of weighted CMP inverse establishing matrix expression for the weighted CMP inverse of an operator and matrix expression for the Moore–Penrose inverse of this new inverse. Applying these results, we introduce and characterize the CMP inverse for a Hilbert space operator.

Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

We investigate the input–output decoupling problem of switched Boolean control networks (SBCNs) in this paper. Based on the matrix expression of Boolean functions, the dynamics of SBCNs are converted into an algebraic form via semi-tensor product of matrices first. Then, using the redundant variable separation technique, we give the necessary and sufficient conditions for the existence of three kinds of controllers to detect whether an SBCN can be input–output decomposed or not, respectively, including the open-loop controllers, the state feedback controllers, and the output feedback controllers. Meanwhile, a constructive procedure is presented to construct the open-loop controllers, as well as the state feedback controllers and output feedback controllers. Finally, an illustrative example is given to show that the new results obtained are effective.

Wind turbine’s shrink disk (WTSD), as a component of transmission system, plays a crucial role in the wind turbine work. In this study, the matrix expression of combined pressure and interference magnitude has been derived by analyzing the geometry relationship of interference fit of multi-layer cylinder. The mechanic model is built by simplifying WTSD as interference fit of multi-layer cylinder. The reliability model is established on the basis of pressure-strength model, and the formula regarding reliability sensitivity is derived. In terms of the minimum principle on the square sum of reliability sensitivity and the minimum reliability principle, the reliability-based robust design (RBRD) model is built under the multi-objective conditions, and the dynamic reliability model is established on the basis of reliability model, load model, and residual strength model. Dynamic reliability sensitivity formula is deduced, and the dynamic reliability-based robust design (DRBRD) model is established under the multi-objective conditions. The effects of the coefficient of friction, assembly clearance, and yield strength on the reliability of WTSD are analyzed by RBRD and DRBRD methods. Taking a type of WTSD as an example, the calculating results validate that the reliability changes along with interference magnitude. The comparison of results shows that the above methods and Monte Carlo method have the same tendency and approximately agree well. Moreover, the traditional design method and RBRD method are compared. It turned out that RBRD can improve the reliability of WTSD by 47.6%, the reliability-based sensitivity is less than that from the traditional design method, and the robustness is stronger. The experiment verified that the design methods of this study can better meet the operation requirements.

This paper studies a class of cooperative games, called graphical cooperative games, where the internal topology of the coalition depends on a prescribed communication graph among players. First, using the semitensor product of matrices, the value function of graphical cooperative games can be expressed as a pseudo-Boolean function. Then, a simple matrix formula is provided to calculate the Shapley value of graphical cooperative games. Finally, some practical examples are presented to illustrate the application of graphical cooperative games in communication-based coalitions and establish the significance of the Shapley value in different communication networks.

In this paper, extremal ranks of some matrix expressions are derived, by using some matrix rank inequalities. We exhibit under suitable conditions a neat relationship between the g-inverse for a sum of two matrices and the g-inverses of the individual terms.

In the Monte Carlo second-order many-body perturbation (MC-MP2) method, the long sum-of-product matrix expression of the MP2 energy, whose literal evaluation may be poorly scalable, is recast into a single high-dimensional integral of functions of electron pair coordinates, which is evaluated by the scalable method of Monte Carlo integration. The sampling efficiency is further accelerated by the redundant-walker algorithm, which allows a maximal reuse of electron pairs. Here, a multitude of graphical processing units (GPUs) offers a uniquely ideal platform to expose multilevel parallelism: fine-grain data-parallelism for the redundant-walker algorithm in which millions of threads compute and share orbital amplitudes on each GPU; coarse-grain instruction-parallelism for near-independent Monte Carlo integrations on many GPUs with few and infrequent interprocessor communications. While the efficiency boost by the redundant-walker algorithm on central processing units (CPUs) grows linearly with the number of electron pairs and tends to saturate when the latter exceeds the number of orbitals, on a GPU it grows quadratically before it increases linearly and then eventually saturates at a much larger number of pairs. This is because the orbital constructions are nearly perfectly parallelized on a GPU and thus completed in a near-constant time regardless of the number of pairs. In consequence, an MC-MP2/cc-pVDZ calculation of a benzene dimer is 2700 times faster on 256 GPUs (using 2048 electron pairs) than on two CPUs, each with 8 cores (which can use only up to 256 pairs effectively). We also numerically determine that the cost to achieve a given relative statistical uncertainty in an MC-MP2 energy increases as O(n

A bivariate rational bicubic interpolating spline (BRIS) with biquadratic denominator and six shape parameters is constructed in a rectangle domain. The C1 continuous condition of BRIS discussed. BRIS is proved to be bounded and its error is estimated. In the case of the equally spaced knots, the matrix expression and symmetry of BRIS are presented. Some properties of the basis of BRIS are given. In the end, a numerical example is given to illustrat the effect of the shape parameters on the shape of BRIS surface.

In [4], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a 2-chain and introduced the notion of chain-walk, which was motivated from graph theory: a directed walk in a directed graph is a sequence of edges with compatible condition on initial and terminal vertices between sequential edges. We consider a directed graph whose vertices are 1-simplices whose supports contain $0$ and edges are plus/minus of $2$-simplices whose supports contain $0$. A chain-walk is a 2-chain induced from a directed walk in this graph. We reduced any 2-chains with 1-shell boundaries into chain-walks having the same boundaries. ¶ In this paper, we reduce any 2-chains of 1-shell boundaries into chain-walks of the same boundary with support of size $3$. Using this reduction, we give a combinatorial criterion determining whether a minimal 2-chain is of RN- or NR-type. For a minimal RN-type 2-chains, we show that it is equivalent to a 2-chain of Lascar type (coming from model theory) if and only if it is equivalent to a planar type 2-chain.

We consider the question of how to takeXsuch that the nonlinear matrix expressionX-XAXattains its maximal and minimal possible ranks.

based on the structural characteristics of the truss structure, the mathematical model design model for a truss structure used in Mast-climbing Work Platform is built with Ansys design language, in which the maximum stress is the optimization objective and the bar orientation is the design variable, showing the constraints and loading condition in mathematical matrix expression, on this premise of the stiffness condition meets the requirement, the bar orientation brings about great influence on the truss structure optimizing, the triangular framework optimized to be quadrangle framework, the maximum stress has been decreased by 74.37%.

This paper addresses the disturbance decoupling problem (DDP) of mix-valued logical networks by using both the semi-tensor product of matrices and the matrix expression of logical functions, and presents a number of new results on the DDP. First, in order to solve the DDP, a new algorithm is proposed to construct a kind of Y-friendly subspace for the state space of mix-valued logical networks. Second, based on the constructed Y-friendly subspace, several necessary and sufficient conditions are obtained for the existence of decoupling controllers, and a new control design procedure is established to deal with the DDP. Finally, an illustrative example is studied by the new method proposed in this paper, and two kinds of decoupling controllers are designed. The study of the illustrative example shows that the new method is effective in dealing with the DDP of mix-valued logical networks.

In this paper, we propose a matrix-based approach for finite automata and then study the reachability conditions. Both the deterministic and nondeterministic automata are expressed in matrix forms, and the necessary and sufficient conditions on reachability are given using semitensor product of matrices. Our results show that the matrix expression provides an effective computational way for the reachability analysis of finite automata.

To constructed turbine blade efficiently and accurately, NURBS is adopted to fit the turbine blade surface. The cross section and longitudinal node vectors are computed using the centripetal parameter method and the accumulation chord length parameter method separately. The accurate mathematics model of blade is built with non-uniform rational B-spline based on control points and node vector. The matrix expression is also given. The example shows the blade mathematics model satisfies the accuracy requirement completely. It has the practical application value to enhance the blade design quality and carry on the blade dynamic analysis.

The extremal ranks of matrix expression of A − BXC with respect to X H = X have been discussed by applying the quotient singular value decomposition Q-SVD and some rank equalities of matrices in this paper.

In order to solve the problem of multidimensional logic variable true value function remarked in the paper(Yaozhi Jiang., 2017), now author has used discrete multiple Fourier transform to deal with the problem remarked above, and obtained an theoretical formulations of discrete multidimensional Fourier transform for that multidimensional logic variable true value function is unknown or we need the frequency properties of multidimensional logic variable true value function. Another problem is about further and deeper properties of Kirchhoff matrices defined in author’s paper(Yaozhi Jiang., 2017), author has established a series of matrix expression for Kirchhoff laws and some new properties of Kirchhoff matrices. These results are all compute-able and complicated.

The matrix Darboux transformation is applied to an auxiliary problem of the classical Wess–Zumino–Novikov–Witten model. One- and two-soliton solutions are written explicitly, and a matrix expression for the N-soliton solution is given. Discrete symmetries of the WZNW model are analyzed, and a solution of the linearized equation of motion is obtained. Bibliography: 19 titles.

The Kronig‐Penney model consists of a periodic potential function, which is used to derive the concepts of allowed and forbidden energy bands. Here, to circumvent the tedious evolution, it is helpful to use computer methods to derive the textbook equation of the analytic Kronig‐Penney model. The hierarchical methodology with the shorter symbols can effectively reduce the complication in the expression of the 4 × 4 matrix. By defining the symbol sequence, we can easily reduce terms by cancellation. With these methods and computer‐aided evolution, we can derive the Kronig‐Penney model from the matrix expression to the textbook form without the extremely laborious evolution.