BPS States, Refined Indices, and Quiver Invariants

For D=4 BPS state construction, counting, and wall-crossing thereof, quiver quantum mechanics offers two alternative approaches, the Coulomb phase and the Higgs phase, which sometimes produce inequivalent counting. The authors have proposed, in arXiv:1205.6511, two conjectures on the precise relationship between the two, with some supporting evidences. Higgs phase ground states are naturally divided into the Intrinsic Higgs sector, which is insensitive to wall-crossings and thus an invariant of quiver, plus a pulled-back ambient cohomology, conjectured to be an one-to-one image of Coulomb phase ground states. In this note, we show that these conjectures hold for all cyclic quivers with Abelian nodes, and further explore angular momentum and R-charge content of individual states. Along the way, we clarify how the protected spin character of BPS states should be computed in the Higgs phase, and further determine the entire Hodge structure of the Higgs phase cohomology. This shows that, while the Coulomb phase states are classified by angular momentum, the Intrinsic Higgs states are classified by R-symmetry.Comment: 51 pages, 5 figure

Randomized block Gram-Schmidt process for solution of linear systems and eigenvalue problems

We propose a block version of the randomized Gram-Schmidt process for computing a QR factorization of a matrix. Our algorithm inherits the major properties of its single-vector analogue from [Balabanov and Grigori, 2020] such as higher efficiency than the classical Gram-Schmidt algorithm and stability of the modified Gram-Schmidt algorithm, which can be refined even further by using multi-precision arithmetic. As in [Balabanov and Grigori, 2020], our algorithm has an advantage of performing standard high-dimensional operations, that define the overall computational cost, with a unit roundoff independent of the dominant dimension of the matrix. This unique feature makes the methodology especially useful for large-scale problems computed on low-precision arithmetic architectures. Block algorithms are advantageous in terms of performance as they are mainly based on cache-friendly matrix-wise operations, and can reduce communication cost in high-performance computing. The block Gram-Schmidt orthogonalization is the key element in the block Arnoldi procedure for the construction of Krylov basis, which in its turn is used in GMRES and Rayleigh-Ritz methods for the solution of linear systems and clustered eigenvalue problems. In this article, we develop randomized versions of these methods, based on the proposed randomized Gram-Schmidt algorithm, and validate them on nontrivial numerical examples

Chebyshev interpolation for nonlinear eigenvalue problems

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a pre-specified curve in the complex plane. A first-order perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element method

structural analysis of transversally loaded quasi isotropic rectilinear orthotropic composite circular plates with galerkin method

Abstract Bending analysis of rectilinear orthotropic composite plates have been scarcely investigated taking into account the increasing use of composite materials in structural applications in the last years. This kind of plates are laminates with axisymmetric geometry and they are made up of unidirectionally reinforced layers with different orientations. Transversally loading this kind of circular plates, the deflected mid-surface is not independent from the circumferential coordinate, unlike the case of isotropic circular plate. Nevertheless, the quasi-isotropic stacking sequence makes still possible to introduce the hypothesis of axisymmetry for the mid-surface deflection under transversal load, disregarding the circumferential variation of the vertical displacement connected to the variable bending stiffness. Then, the constitutive equations for this specific family of plates were obtained finding the stress resultants-strains relations in the global cylindrical coordinate system. These expressions, along with the equilibrium equations, made it possible to derive the governing equation of the problem in the frame of Kirchhoff-Love hypothesis of the classical lamination theory. The Galerkin method was applied to solve the governing third order differential equation in terms of mid-surface deflection, introducing appropriate polynomial approximation functions compliant with the boundary conditions. In particular, fully clamped constraint conditions were considered for the outer diameter of the plate in conjunction with an internal rigid core. The characterization of this model allows to define the stiffness matrix terms of a custom composite bolted joint finite element, that is the object of future developments of this work. Results of the original proposed method are presented and compared to those obtained by means of FEA performed with a refined reference model, demonstrating a good agreement