A new, globally convergent Riemannian conjugate gradient method

This article deals with the conjugate gradient method on a Riemannian manifold with interest in global convergence analysis. The existing conjugate gradient algorithms on a manifold endowed with a vector transport need the assumption that the vector transport does not increase the norm of tangent vectors, in order to confirm that generated sequences have a global convergence property. In this article, the notion of a scaled vector transport is introduced to improve the algorithm so that the generated sequences may have a global convergence property under a relaxed assumption. In the proposed algorithm, the transported vector is rescaled in case its norm has increased during the transport. The global convergence is theoretically proved and numerically observed with examples. In fact, numerical experiments show that there exist minimization problems for which the existing algorithm generates divergent sequences, but the proposed algorithm generates convergent sequences.Comment: 22 pages, 8 figure

Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models

The redistribution of energy levels between energy bands is studied for a family of simple effective Hamiltonians depending on one control parameter and possessing axial symmetry and energy-reflection symmetry. Further study is made on the topological phase transition in the corresponding semi-quantum and completely classical models, and finally the joint spectrum of the two commuting observables $(H=E,J_z)$ (also called the lattice of quantum states) is superposed on the image of the energy-momentum map for the classical model. Through these comparative analyses, mutual correspondence is demonstrated to exist among the redistribution of energy levels between energy bands for the quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian monodromy for the complete classical analog. In particular, as far as the band rearrangement is concerned, a fine agreement is found between the redistribution of the energy levels described in terms of joint spectrum of energy and momentum in the full quantum model and the evolution of singularities of the energy-momentum map of the complete classical model. The topological phase transition observed in the present semi-quantum and the complete classical models are analogous to topological phase transitions of matter

The 2D Kramers-Dirac oscillator and a corresponding semi-quantum system (Symmetry and Singularity of Geometric Structures and Differential Equations)

The 2D Kramers-Dirac oscillator and its corresponding semi-quantum Hamiltonian are introduced. The bulk-edge correspondence is shown to hold in terms of spectral flow and Chern number

A Note on Canonical Transforms Representing SL₂ (2, R), a Two-Fold Covering of SL (2, R), in L² (R)

Canonical transforms representing unitarily SL₂(2, R), a two-fold covering of SL(2, R), are explicitly constructed in the form of integral transforms in L²(R) with the main stress laid on the composition of canonical transforms

Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics

This paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifold M admitting a compact Lie group G as an isometry group, the quotient space Q = M/G is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over Q = M/G. This idea of reduction fails, if the action of G on M is not free. However, the Peter-Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter-Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the stratified manifold M. Then the reduced Laplacian is well defined as a self-adjoint operator with the boundary conditions on singular sets of lower dimensions. Application to quantum molecular mechanics is also discussed in detail. In fact, the reduction of quantum systems studied in this paper stems from molecular mechanics. If one wishes to consider the molecule which is allowed to lie in a line when it is in motion, the reduction method presented in this paper works well.Comment: 33 pages, no figure

Activation of fibroblast-like synoviocytes derived from rheumatoid arthritis via lysophosphatidic acid-lysophosphatidic acid receptor 1 cascade

INTRODUCTION: Lysophosphatidic acid (LPA) is a bioactive lipid that binds to G protein–coupled receptors (LPA(1–6)). Recently, we reported that abrogation of LPA receptor 1 (LPA(1)) ameliorated murine collagen-induced arthritis, probably via inhibition of inflammatory cell migration, Th17 differentiation and osteoclastogenesis. In this study, we examined the importance of the LPA–LPA(1) axis in cell proliferation, cytokine/chemokine production and lymphocyte transmigration in fibroblast-like synoviocytes (FLSs) obtained from the synovial tissues of rheumatoid arthritis (RA) patients. METHODS: FLSs were prepared from synovial tissues of RA patients. Expression of LPA(1–6) was examined by quantitative real-time RT-PCR. Cell surface LPA(1) expression was analyzed by flow cytometry. Cell proliferation was analyzed using a cell-counting kit. Production of interleukin 6 (IL-6), vascular endothelial growth factor (VEGF), chemokine (C-C motif) ligand 2 (CCL2), metalloproteinase 3 (MMP-3) and chemokine (C-X-C motif) ligand 12 (CXCL12) was measured by enzyme-linked immunosorbent assay. Pseudoemperipolesis was evaluated using a coculture of RA FLSs and T or B cells. Cell motility was examined by scrape motility assay. Expression of adhesion molecules was determined by flow cytometry. RESULTS: The expression of LPA(1) mRNA and cell surface LPA(1) was higher in RA FLSs than in FLSs from osteoarthritis tissue. Stimulation with LPA enhanced the proliferation of RA FLSs and the production of IL-6, VEGF, CCL2 and MMP-3 by FLSs, which were suppressed by an LPA(1) inhibitor (LA-01). Ki16425, another LPA(1) antagonist, also suppressed IL-6 production by LPA-stimulated RA FLSs. However, the production of CXCL12 was not altered by stimulation with LPA. LPA induced the pseudoemperipolesis of T and B cells cocultured with RA FLSs, which was suppressed by LPA(1) inhibition. In addition, LPA enhanced the migration of RA FLSs and expression of vascular cell adhesion molecule and intercellular adhesion molecule on RA FLSs, which were also inhibited by an LPA(1) antagonist. CONCLUSIONS: Collectively, these results indicate that LPA–LPA(1) signaling contributes to the activation of RA FLSs