Null Matrix Research Articles

Efficient finite element analysis using graph-theoretical force method; rectangular plane stress and plane strain serendipity family elements

Formation of a suitable null basis for equilibrium matrix is the main part of finite elements analysis via force method. Foran optimal analysis, the selected null basis matrices should be sparse and banded corresponding to produce sparse, banded and well-conditioned flexibility matrices. In this paper, an efficient method is developed for the formation of null bases of finite element models (FEMs) consisting of rectangular plane stress and plane strain serendipity family elements, corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs with the FEM and selecting appropriate subgraphs and forming the self-equilibrating systems (SESs) on these subgraphs. The efficiency of the present method is illustrated through three examples.

An Algebraic Analysis of the Graph Modularity

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank-one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency and Laplacian matrices. In this paper we analyze certain spectral properties of modularity matrices, which are related to the community detection problem. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between the graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph modularity.

An analysis dictionary learning algorithm under a noisy data model with orthogonality constraint.

Two common problems are often encountered in analysis dictionary learning (ADL) algorithms. The first one is that the original clean signals for learning the dictionary are assumed to be known, which otherwise need to be estimated from noisy measurements. This, however, renders a computationally slow optimization process and potentially unreliable estimation (if the noise level is high), as represented by the Analysis K-SVD (AK-SVD) algorithm. The other problem is the trivial solution to the dictionary, for example, the null dictionary matrix that may be given by a dictionary learning algorithm, as discussed in the learning overcomplete sparsifying transform (LOST) algorithm. Here we propose a novel optimization model and an iterative algorithm to learn the analysis dictionary, where we directly employ the observed data to compute the approximate analysis sparse representation of the original signals (leading to a fast optimization procedure) and enforce an orthogonality constraint on the optimization criterion to avoid the trivial solutions. Experiments demonstrate the competitive performance of the proposed algorithm as compared with three baselines, namely, the AK-SVD, LOST, and NAAOLA algorithms.

Distributed Operational Space Formulation of Serial Manipulators

This paper presents a new parallel algorithm for the operational space dynamics of unconstrained serial manipulators, which outperforms contemporary sequential and parallel algorithms in the presence of two or more processors. The method employs a hybrid divide and conquer algorithm (DCA) multibody methodology which brings together the best features of the DCA and fast sequential techniques. The method achieves a logarithmic time complexity (O(log(n)) in the number of degrees of freedom (n) for computing the operational space inertia (Λe) of a serial manipulator in presence of O(n) processors. The paper also addresses the efficient sequential and parallel computation of the dynamically consistent generalized inverse (J¯e) of the task Jacobian, the associated null space projection matrix (Ne), and the joint actuator forces (τnull) which only affect the manipulator posture. The sequential algorithms for computing J¯e, Ne, and τnull are of O(n), O(n2), and O(n) computational complexity, respectively, while the corresponding parallel algorithms are of O(log(n)), O(n), and O(log(n)) time complexity in the presence of O(n) processors.

Spectral density of a Wishart model for nonsymmetric correlation matrices

The Wishart model for real symmetric correlation matrices is defined as W=AA^{t}, where matrix A is usually a rectangular Gaussian random matrix and A^{t} is the transpose of A. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices A and B as AB^{t}. The corresponding Wishart model, thus, is defined as C=AB^{t}BA^{t}. We study the spectral density of C for the case when A and B are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of C at large matrix dimension. We complement our analytic results with numerics.

Efficient finite element analysis using graph-theoretical force method; hexahedron elements

Formation of a suitable null basis for equilibrium matrix is the main problem of finite elements analysis via force method. For an optimal analysis, the selected null basis matrices should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrices. In this paper, an efficient method is developed for the formation of null bases of finite element models (FEMs) consisting of hexahedron elements, corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs with the FEM and selecting appropriate subgraphs and forming the self-equilibrating systems (SESs) on these subgraphs.

Hybrid textures of the right-handed Majorana neutrino mass matrix

We perform a systematic study of neutrino mass matrices having a vanishing cofactor and an equality between two cofactors of the mass matrix. Such texture structures of the effective neutrino mass matrix arise from type-I seesaw mechanism when the Dirac neutrino mass matrix is diagonal with equal elements and the right-handed Majorana neutrino mass matrix has hybrid textures with one equality of matrix elements and one zero matrix element. For three right-handed neutrinos there are sixty possible hybrid textures out of which only six are excluded by the present experimental data. We show that such textures can be derived using discrete symmetries. The predictions of experimentally allowed textures are examined for unknown parameters such as the effective Majorana mass of the electron neutrino and the Dirac-type CP-violating phase.

Distributed Sum-Rate Optimization for Full-Duplex MIMO System Under Limited Dynamic Range

Distributed sum-rate-maximizing covariance matrices design for full-duplex multi-input multi-output communication is considered, where the information of the loopback interference channels cannot be exchanged reliably due to their large dynamic ranges. We propose a structured covariance matrices design which finds the optimal balance between the two solutions in the extremes of the weak and strong self-interference through a single-parameter optimization. We further propose a low-complexity null projection matrix design algorithm, in which the solution in the strong self-interference regime is designed in the sense to maximize the received channel gain. Exploiting the well-posed structure, the proposed scheme nearly achieves the sum-rate of the previous scheme based on the gradient projection with significantly less amount of inter-node iterations, yielding an increased effective sum-rate and reduced overall complexity for the practial channel block lengths.

The Kronecker Join of the Boolean Matrices

In this paper, we define the Kronecker join of the Boolean matrices, and discuss its properties and show that the index lattice of the Kronecker join of the Boolean matrices is isomorphic to the tensor product of the index lattices of the factor Boolean matrices. First, we introduce the index lattice of the Boolean matrix. Second, we define the Kronecker join of the Boolean matrices and discuss its properties. Finally, we show that the index lattice of the Kronecker join is isomorphic to the tensor product of the index lattices of the factor Boolean matrices. Boolean matrix has entries that are 0 or 1. Usually, we denote the sets of the Boolean matrices with m rows and n columns by βmn .A matrixA is a null-matrix if all the elements are 0 and a universe-matrix when all the elements are 1 which is denoted by E.L etA =( aij )m×n, B =( bij)m×n .T hei-row and j-column of A are denoted by Ai∗ and A∗j respectively. We say A ≤ B if aij ≤ bij ,i =1 , 2, ··· ,m and j =1 , 2, ··· ,n and we say A<B if A ≤ B and AB. More detail information about Boolean matrix and lattice theory can be found in (5-6).

The syntax of Japanese tokoro-clauses: Against control analyses

The Japanese tokoro-clause construction has long been an intriguing topic in Japanese linguistics because of its mismatch between syntax and semantics. Syntactically, the tokoro-clause itself seems like an object of the matrix verb through being assigned an accusative or a dative particle, while semantically the subject of the tokoro-clause is construed as the thematic object of the matrix verb. It has been argued that this mismatch can be reconciled by assuming backward object control in which the unpronounced matrix object is coreferential with the downstairs tokoro-clause subject: NP eci [NPi V tokoro]-acc/dat V. While acknowledging four different types of tokoro-clauses according to the particle and the matrix verb, it is argued that none of them involve obligatory control, let alone backward control. Specifically, two of them are complements on their own, and the other two are vP-adjuncts. Although there is a null matrix object in the latter case, it does not have a c-command relation with the tokoro-clause subject, and its referential relation is pronominal coreference rather than obligatory control.

Further improvement in stability criteria for linear systems with interval time‐varying delay

The delay-dependent stability problem for systems with time-delay varying in an interval is addressed in this study. Based on Lyapunov–Krasovskii theory the proposed methods formulated as linear matrix inequality problems are able to check the stability interval when the time-varying delay d(t) belongs to an interval [τ1, τ2]. The Lyapunov–Krasovskii functional (LKF) selected in the present paper is simpler than some ones considered in the literature. However, the criteria obtained, based on this simple LKF, outperformed the similar existing ones in all numerical tests accomplished in this paper. Notation: Through out this paper the superscript T stands for transpose. 0 refers to a null matrix with appropriate dimensions. For a real symmetric matrix M, the notation M > 0 ( < 0) means that M is the positive (negative) definite. The symmetric term in a matrix is denoted by *.

A null model algorithm for presence–absence matrices based on proportional resampling

Ecological presence–absence matrices capture information of species occurrences among a number of sites. Statistical inference of matrix structure often used a fixed–fixed (FF) null model in which matrix entries are randomized, but the row and column total of each random matrix match those of the original matrix. However, in a stochastically assembled meta-community, row and column totals of a random assemblage might be expected to vary among matrices. Here we introduce a 4-step proportional–proportional (PP) algorithm that creates null matrices in which the row and column vary randomly, but the average row and column totals in a set of PP matrices are unbiased and match those of the original matrix. We tested the performance of the PP algorithm with 5 sets of artificial matrices and one large set of 288 published empirical matrices. Compared to the FF algorithm, the PP algorithm has better power to detect segregated and nested matrices, but it is vulnerable to Type I errors if row and column sums have small variances. The PP algorithm identified only 9% of empirical matrices as significantly segregated, compared with 30% identified by the traditional FF algorithm. The choice between whether to use the PP or the FF algorithm is similar to the distinction between random and fixed effects in a mixed-model ANOVA. For robust analysis, it may be desirable to use both the PP and the FF algorithms with the same data matrix.

Simple disjunctive normal forms of Boolean functions with a restricted number of zeros

This paper studies the complexity of disjunctive normal forms (DNFs) of Boolean functions defined by their matrices of zeros. The best available bounds are obtained for the minimum number of conjunc� tions making up a DNF of a Boolean function with a restricted number of zeros. The results can be used to implement characteristic functions of classes in pat� tern recognition with binary data [1]. 1. The standard method for constructing DNFs of characteristic classes of functions can be described as follows. Let , , …, and , , …, be given binary descriptions of reference objects of classes K1 and K2, respectively, where , i ∈ {0, 1}, j ∈ {1, 2, …, ki}, is a binary vector of dimension n.

Discrete Null Space Method for the Newmark Integration of Multibody Dynamic Systems

Considering that the algorithm of the equations of motion of multibody systems is the base for the analysis of dynamics,the discrete null space method for the Newmark integration is proposed.The procedure is essentially based upon the following steps.The equations of motion of constrained multibody systems are directly discretized by the Newmark time stepping scheme.The discrete Lagrange multipliers are eliminated by using a discrete null space matrix and the discrete d'Alembert-type formulation is obtained.The number of unknowns is reduced further by employing specific local reparametrizations.It is shown that the developed method can not only reduce the dimension of the equations of motion,thus improving the computational efficiency,but also obtain the good results with less constraint violation.

Singularity-free simulation of closed loop multibody systems by using null space of Jacobian matrix

Numerical simulation of closed loop multibody systems is associated with numerical solution of equations of motion which are, in general, in the form of DAE’s index-3 systems. For assuring continuous simulation, one should overcome some difficulties such as stabilization of the constraint equations, singular configuration of the system. In this paper, the system equations of motion with the Lagrange multipliers is rewritten by introducing generalized reaction forces. The combination with the condition of ideality of constraints leads to the system of equations which can be solved by numerical techniques smoothly, even over singular positions. Based on the new criterion of ideality of constraints, which relates generalized reaction forces and the null space matrix of Jacobian matrix, it is possible also to remove reaction forces and use only the reduced system of equations with null space matrix for passing singular positions. In order to prevent the constraint equations from the accumulated errors of integral time, the method of position and velocity projection has been exploited. Some numerical experiments are carried out to verify the proposed approach.

On iterative methods for the quadratic matrix equation with M-matrix

In this paper, we consider Newton’s method and Bernoulli’s method for a quadratic matrix equation arising from an overdamped vibrating system. By introducing M-matrix to this equation, we provide a sufficient condition for the existence of the primary solution. Moreover, we show that Newton’s method and Bernoulli’s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition.

Efficient finite element analysis by graph-theoretical force method; rectangular and triangular plate bending elements

The calculation of null basis for equilibrium matrix is the main part of the finite elements analysis via force method. For an optimal analysis, the selected null basis matrices should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrices. There are many algorithms for the formation of null basis among which the algebraic methods benefit from the generality. However, the efficiency of these methods is highly dependent on the size of problems, and their computational times are very high for such problems. In this paper, a graph-theoretical method is presented for the formation of sparse, banded and highly accurate null basis matrices for finite element models with triangular and rectangular bending elements. These bases are generated much faster than those obtained by the algebraic methods. The efficiency of the present method is illustrated through some example.

Response to Collins et al. (2011)

We note serious problems in Collins et al. (Journal of Biogeography, 2011, doi: 10.1111/j.1365-2699.2011.02506.x): failure to use over 80% of the available data; failure to use one of the two available archipelagoes; mistaken inclusion of four species; and reliance on a grossly inadequate number of null matrices. Curing the paper of these problems would have strengthened the evidence for checkerboards and the role of competition.

Integrable coupling hierarchy and Hamiltonian structure for a matrix spectral problem with arbitrary-order

We presented an integrable coupling hierarchy of a matrix spectral problem with arbitrary order zero matrix r by using semi-direct sums of matrix Lie algebra. The Hamiltonian structure of the resulting integrable couplings hierarchy is established by means of the component trace identities. As an example, when r is 2 × 2 zero matrix specially, the integrable coupling hierarchy and its Hamiltonian structure of the matrix spectral problem are computed.

A ternary-arithmetic topological based algebraic method for networks traffic observability

In this paper an algebraic method, which shares all the advantages of the topological methods and allows us to obtain the same results as the standard algebraic method with a substantial reduction in memory and cpu requirements, is presented. The main idea consists of writing the link, OD and scanned flows in terms of route instead of OD flows. This alternative permits starting the algebraic and topological processes with identical matrices of zeros and ones. In addition, in most iterations the pivots can be selected in such a way that the resulting matrices after each iteration contain only zeroes, ones and minus ones. This allows us to design a ternary arithmetic which reproduces the algebraic results exactly, requires only two bits to store each matrix entry and have no precision or non-zero pivot identification problems. Only when this process cannot be continued, the pure algebraic method is used, but only with a very reduced size matrix when compared with the size of the initial matrix. The method is illustrated by its application to a very simple network and to a real network example (the city of Cuenca, Spain).