Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Parallel SVD Algorithm for a Three-Diagonal Matrix on a Video Card Using the Nvidia CUDA Architecture

SVD (Singular Value Decomposition) algorithm is used in recommendation systems, machine learning, image processing, and in various algorithms for working with matrices which can be very large and Big Data, so, given the peculiarities of this algorithm, it can be performed on a large number of computing threads that have only video cards.CUDA is a parallel computing platform and application programming interface model created by Nvidia. It allows software developers and software engineers to use a CUDA-enabled graphics processing unit for general purpose processing – an approach termed GPGPU (general-purpose computing on graphics processing units). The GPU provides much higher instruction throughput and memory bandwidth than the CPU within a similar price and power envelope. Many applications leverage these higher capabilities to run faster on the GPU than on the CPU. Other computing devices, like FPGAs, are also very energy efficient, but they offer much less programming flexibility than GPUs.The developed modification uses the CUDA architecture, which is intended for a large number of simultaneous calculations, which allows to quickly process matrices of very large sizes. The algorithm of parallel SVD for a three-diagonal matrix based on the Givents rotation provides a high accuracy of calculations. Also the algorithm has a number of optimizations to work with memory and multiplication algorithms that can significantly reduce the computation time discarding empty iterations.This article proposes an approach that will reduce the computation time and, consequently, resources and costs. The developed algorithm can be used with the help of a simple and convenient API in C ++ and Java, as well as will be improved by using dynamic parallelism or parallelization of multiplication operations. Also the obtained results can be used by other developers for comparison, as all conditions of the research are described in detail, and the code is in free access.

On the Ger\v{s}gorin disks of distance matrices of graphs

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

Laplacian integral graphs with a given degree sequence constraint

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

On the Sum of Distance Laplacian Eigenvalues of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.

Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

On the derivative-free quasi-Newton-type algorithm for separable systems of nonlinear equations

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

Type Synthesis of Fully Decoupled Three Translational Parallel Mechanism with Closed loop Units and High Stiffness

In this paper, a new synthesis method of fully decoupled three translational (3T) parallel mechanisms (PMs) with closed loop units and high stiffness is proposed based on screw theory. Firstly, a new criterion for the full decoupled of PMs is presented that the reciprocal product of the transmission wrenc h screw matrix and the output twist screw matrix of PMs is a diagonal matrix, and all elements on the main diagonal are nonzero constants. The forms of the transmission wrench screws are determined by the criterion. Secondly, the forms of the actuated and unactuated screws can be obtained according to their relationships with the transmission wrench screws. The basic decoupled limbs are generated by combination of the above actuated and unactuated screws. Finally, a closed loop units construction method is investigated to apply the decoupled mechanisms in a better way on the high stiffness occasion. The closed loop units are constructed in the basic decoupled limbs to generate a high stiffness fully decoupled 3T PM. Kinematic and stiffness analys e s show that the Jacobian matrix is a diagonal matrix, and the stiffness is obviously higher than that of the orthogonal coupling mechanisms, which verifies the correctness of the proposed synthesis method. The mechanism synthesized by this method has a good applicati on prospect in vehicle durability test platform.