Nonsymmetric Case Research Articles

Adaptive rational Krylov subspaces for large-scale dynamical systems

The rational Krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems. However, its success has been hindered by the lack of a parameter-free procedure, which would effectively generate the sequence of shifts used to build the space. In this paper we propose an adaptive computation of these shifts. The whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix, while further information is automatically generated during the process. The approach is a full generalization to the nonsymmetric case of the idea first proposed in Druskin et al. (2010) [18] and it is used for two important problems in control: the approximation of the transfer function and the numerical solution of large Lyapunov equations. The procedure can be naturally extended to other related problems, such as the solution of the Sylvester equation, and parametric or higher order systems. Several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors.

Analysis of Thermo-Mechanical Response in an Aircraft Piston Engine by Analytical, FEM, and Test-Stand Investigations

In the paper, a 3D model of the cylinder sleeve and head as well as their assembly are analyzed in terms of thermo-mechanical loadings. In the initial study, analytical expressions for stresses and deformations in the cylinder-head connection are set forth and examined. In that area, wear of the cylinder wall is most intensive and strongly non-uniform. The proposed model gives satisfactory results for the axially non-symmetric case. Computer simulations on a FE model confirm analytical predictions. That constitutes a basis for formulation of some structural modifications to the considered cylinder system to reduce the non-uniformity of cylinder wall deformations.

Homogeneous paracontact metric three-manifolds

The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.

BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space

In this Note we introduce BV functions in a Gelfand triple, which is an extension of BV functions in Ambrosio et al., preprint [1], by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set Γ. We prove the existence and uniqueness of a strong solution of this problem when Γ is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when Γ = K α , where K α = { f ∈ L 2 ( 0 , 1 ) | f ⩾ − α } , α ⩾ 0 .

Charge orbits of extremal black holes in five-dimensional supergravity

We derive the $U$-duality charge orbits, as well as the related moduli spaces, of ``large'' and ``small'' extremal black holes in nonmaximal ungauged Maxwell-Einstein supergravities with symmetric scalar manifolds in $d=5$ space-time dimensions. The stabilizer groups of the various classes of orbits are obtained by determining and solving suitable $U$-invariant sets of constraints, both in ``bare'' and ``dressed'' charge bases, with various methods. After a general treatment of attractors in real special geometry (also considering nonsymmetric cases), the $\mathcal{N}=2$ ``magic'' theories, as well as the $\mathcal{N}=2$ Jordan symmetric sequence, are analyzed in detail. Finally, the half-maximal ($\mathcal{N}=4$) matter-coupled supergravity is also studied in this context.

Discrete General Lotto Games - The Non-Symmetric Case

We provide a complete characterization of optimal strategies for both players in non-symmetric discrete General Lotto games, where one of the players has an advantage over the other. By this we complete the characterization given in Hart (2008), where the strategies for symmetric case were fully characterized and some of the optimal strategies for the non-symmetric case were obtained. Our results can be used to solve the remaining cases of Colonel Blotto games.

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.

Binomial Coefficients and Littlewood-Richardson Coefficients for Jack Polynomials

In this paper, we consider translation and multiplication operators acting on the rings of symmetric and nonsymmetric polynomials and study their matrix coefficients with respect to the bases of Jack polynomials and interpolation polynomials. The main new insight is that the symmetric and nonsymmetric cases share a key combinatorial feature, that of a locally finite graded poset with a minimum element. This allows us to treat both cases in a simple and unified manner.

Fast transforms for high order boundary conditions in deconvolution problems

We study strategies to increase the precision in deconvolution models, while maintaining the complexity of the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues, etc.) of the same order of the celebrated fast Fourier transform. The key idea is the choice of a suitable functional basis to represent signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in Serra Capizzano (SIAM J. Sci. Comput. 25(3):1307–1325, 2003), we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case.

Infinitely many solutions for symmetric and non-symmetric elliptic systems

We study an elliptic system equivalent to a fourth order elliptic equation. By using variational and perturbative methods, we prove the existence of infinitely many solutions both in the symmetric and in the non-symmetric case.

Algebraic analysis of two‐grid methods: The nonsymmetric case

AbstractTwo‐grid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are provided that characterize all eigenvalues of the iteration matrix. In the symmetric positive‐definite (SPD) case, these expressions reproduce the sharp two‐grid convergence estimate obtained by Falgout, Vassilevski and Zikatanov (Numer. Linear Algebra Appl. 2005; 12:471–494), and also previous algebraic bounds, which can be seen as corollaries of this estimate. These results allow to measure the convergence by checking ‘approximation properties’. In this work, proper extensions of the latter to the nonsymmetric case are presented. Sometimes approximation properties for the SPD case are summarized in loose terms; e.g.: Interpolation must be able to approximate an eigenvector with error bound proportional to the size of the eigenvalue (SIAM J. Sci. Comp. 2000; 22:1570–1592). It is shown that this can be applied to nonsymmetric problems too, understanding ‘size’ as ‘modulus’. Eventually, an analysis is developed, for the nonsymmetric case, of the theoretical foundations of ‘compatible relaxation’, according to which a Fine/Coarse partitioning may be checked and possibly improved. Copyright © 2009 John Wiley & Sons, Ltd.

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n x n symbols, consider its submatrices of size k x k, obtained by deleting n―k rows and n―k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace the multiset by the sum of submatrices. For k > cn 2/3 we prove that the matrix A is determined by the sum of the k x k submatrices, both in the symmetric and in the nonsymmetric cases.

The Result of Two Steps of the $LR$ Algorithm is Diagonally Similar to the Result of One Step of the $HR$ Algorithm

Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the $QR$ algorithm is used, but this destroys tridiagonal form by converting the matrix to Hessenberg form, resulting in increased storage requirements and numerical operations. The $HR$ algorithm, based on the $HR$ factorization of the matrix into a $(\Delta,\Delta_1)$-orthogonal part H, where $H^T\Delta H=\Delta_1$, and an upper triangular part R, solves this problem. In a result proved by Hongguo Xu, two steps of the $LR$ algorithm are equivalent to one step of the $QR$ algorithm for symmetric matrices. The first object of this paper is to use the $HR$ algorithm to extend Hongguo Xu's result to the nonsymmetric case. Since an $HR$ factorization does not always exist, so we also consider an extension to it called $XHR$ factorization. We then prove a similar result about it.

Construction of topological defect networks with complex scalar fields

This work deals with the construction of networks of topological defects in models described by a single complex scalar field. We take advantage of the deformation procedure recently used to describe kinklike defects in order to build networks of topological defects, which appear from complex field models with potentials that engender a finite number of isolated minima, both in the case where the minima present discrete symmetry, and in the non-symmetric case. We show that the presence of symmetry guide us to the construction of regular networks, while the non-symmetric case gives rise to irregular networks which spread throughout the complex field space. We also discuss bifurcation, a phenomenon that appear in the non-symmetric case, but is washed out by the deformation procedure used in the present work.

Dimensional reduction as a method to obtain dual theories for massive spin two in arbitrary dimensions

Using the parent Lagrangian method together with a dimensional reduction from $D$ to $(D\ensuremath{-}1)$ dimensions, we construct dual theories for massive spin two fields in arbitrary dimensions in terms of a mixed symmetry tensor ${T}_{A[{A}_{1}{A}_{2}\dots{}{A}_{D\ensuremath{-}2}]}$. Our starting point is the well-studied massless parent action in dimension $D$. The resulting massive Stueckelberg-like parent actions in $(D\ensuremath{-}1)$ dimensions inherit all the gauge symmetries of the original massless action and can be gauge fixed in two alternative ways, yielding the possibility of having a parent action with either a symmetric or a nonsymmetric Fierz-Pauli field ${e}_{AB}$. Even though the dual sector in terms of the standard spin two field includes only the symmetrical part ${e}_{{AB}}$ in both cases, these two possibilities yield different results in terms of the alternative dual field ${T}_{A[{A}_{1}{A}_{2}\dots{}{A}_{D\ensuremath{-}2}]}$. In particular, the nonsymmetric case reproduces the Freund-Curtright action as the dual to the massive spin two field action in four dimensions.

Adaptive Tracking Controller Design of Nonlinear Systems With Time Delays and Unknown Dead-Zone Input

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper focuses on the tracking control problem for a class of nonlinear system with time delays and dead-zone input. The nonsymmetric dead-zone input case is considered without the knowledge of the dead zone parameters. The time delay uncertainties are bounded by a nonlinear function with unknown coefficients. By constructing a novel Lyapunov functional, we design a simple and smooth adaptive state feedback controller. It is shown that the solution of the resulting closed-loop error system converges to an adjustable region exponentially. Finally, numerical examples are included to show the effectiveness of the theoretical results. </para>

Left conjugate gradient method for non‐Hermitian linear systems

AbstractRecently, Yuan et al. (BIT: Numer. Math. 2004; 44(1):189–207) proposed the left conjugate gradient (LCG) method for real positive‐definite linear systems. This paper aims to generalize their method for solving complex non‐Hermitian linear systems. To avoid the breakdown that possibly occurred in the LCG method, we also propose the block left conjugate direction method and the block LCG (BLCG) method. It is found that no breakdown occurs in the BLCG method and the block idea also applies to the real nonsymmetric case. Numerical experiments demonstrate the usefulness of the proposed LCG method. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Scalar curvature equation on [formula omitted], Part I: Topological conditions

This is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension. In the first part, we will adopt the degree-theoretic approach to give a topological condition and some general, explicit conditions on the scalar curvature functions to ensure the solvability of the problem. Our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function, which is derived from the asymptotic expansion of the boundary map introduced in [A. Chang, P. Yang, A perturbation result in prescribing scalar curvature on S n , Duke Math. J. 64 (1991) 27–69]. Our conditions, particularly allowing non-isolation and non-degeneracy of the critical points of the scalar curvature functions, can be easily verified in many situations. In the second part of series, we will make a detailed study on the verification of the topological condition. Our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non-symmetric cases of the scalar curvature functions.

ON THE MARGIN OF COMPLETE STABILITY FOR A CLASS OF CELLULAR NEURAL NETWORKS

In this paper, the dynamical behavior of a class of third-order competitive cellular neural networks (CNNs) depending on two parameters, is studied. The class contains a one-parameter family of symmetric CNNs, which are known to be completely stable. The main result is that it is a generic property within the family of symmetric CNNs that complete stability is robust with respect to (small) nonsymmetric perturbations of the neuron interconnections. The paper also gives an exact evaluation of the complete stability margin of each symmetric CNN via the characterization of the whole region in the two-dimensional parameter space where the CNNs turn out to be completely stable. The results are established by means of a new technique to investigate trajectory convergence of the considered class of CNNs in the nonsymmetric case.

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ1(w), . . . , Δr(w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δk(w)/Δk-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.