On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

On the ricci tensor of a real hypersurface of quaternionic hyperbolic space

Summary We prove the non-existence of Einstein real hypersurfaces of quaternionic hyperbolic space. This article was processed by the author using the IAT_EX style filecljour1 from Springer-Verlag.     

Transformations of linear connections,II

We elucidate [9] with two applications. In the first we view connections as differential systems. Specializing this to trivial bundles overS ¹ and applying the theory of Floquet, we obtain equivalent connections with constant Christoffel symbols. In the second application we prove that the canonical connections of parallelizable manifolds (in particular Lie groups) can be obtained from the canonical flat connection of appropriate trivial bundles. Thus, the formalisms of [1], [4], [5] and [6] fit in the general setting of [9].     

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of \mathbbZ/2{\mathbb{Z}/2} by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.     

On some subclasses of P‐matrices

A matrix with positive row sums and all its off‐diagonal elements bounded above by their corresponding row averages is called a B‐matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22 :1027–1037) and (Numer. Math. 2003; 95 :337–345). In this paper, it is generalized to more extended matrices—MB‐matrices, which is proved to be a subclass of the class of P‐matrices. Subsequently, we establish relationships between defined and some already known subclasses of P‐matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21 :67–78; Linear Algebra Appl. 2004; 393 :353–364; Linear Algebra Appl. 1995; 225 :117–123). As an application, some subclasses of P‐matrices are used to localize the real eigenvalues of a real matrix. Copyright © 2007 John Wiley & Sons, Ltd.     

On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric

In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kähler-Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kähler-Finsler metric F is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type (2,0) of the complex Berwald connection vanish; (iii) the complex Wrona metric in Cn is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S2n−1⊂Cn is explicitly obtained.     

On some properties of the curvature and Ricci tensors in complex affine geometry

We present a new approach — which is more general than the previous ones — to the affine differential geometry of complex hypersurfaces inC ⁿ⁺¹. Using this general approach we study some curvature conditions for induced connections.The research supported by Alexander von Humboldt Stiftung and KBN grant no. 2 P30103004.     

Planar normal sections on the natural imbedding of a flag manifold

We study the setX[M] of planar normal sections on the natural imbedding of a flag manifoldM. We characterizeX[M] and show that it is a real algebraic submanifold of P ⁿ (n=dimM). From results of Chen and Ferus, it may be concluded thatX[M] measures how farM is from a symmetricR-space. We compute the Euler characteristic ofX[M] and its complexificationX _c [M]. Our main result shows, in particular, thatx(X[M]) andx(X _c [M]) depend only on dimM and not on the nature ofM itself.     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

Geodesics on real hypersurfaces of type A 2 in a complex space form

In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A ₂ in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the ambient complex space form. Author’s address: Toshiaki Adachi, Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan     

Bundle automorphisms for fiberG-structures of finite type

In this paper we study geometric settings where a Lie group preserving a measurable field of measurable Riemannian metrics on the fibers of a smooth fiber bundle must actually preserve a measurable field of smooth Riemannian metrics. For ergodic actions on bundles with compact fiber this will imply that the standard fiber is a homogeneous space for a compact Lie group. In particular we show this conclusion holds for a semisimple Lie group of higher real rank (or a lattice subgroup) preserving a finite measure and either a field of connections or pseudo-Riemannian metrics when the fiber is compact and of low dimension.Research completed while a member of the University of Chicago Mathematics Department.     

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.     

Partial characterizations of circular‐arc graphs

A circular‐arc graph is the intersection graph of a family of arcs on a circle. A characterization by forbidden induced subgraphs for this class of graphs is not known, and in this work we present a partial result in this direction. We characterize circular‐arc graphs by a list of minimal forbidden induced subgraphs when the graph belongs to any of the following classes: P₄ ‐free graphs, paw‐free graphs, claw‐free chordal graphs and diamond‐free graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 289–306, 2009     

On the Curvature Groups of a CR Manifold

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups of the Fefferman space C(M) of a strictly pseudoconvex real hypersurface Dedicated to the memory of Professor Aldo Cossu The Authors acknowledge support from INdAM (Italy) within the interdisciplinary project Nonlinear subelliptic equations of variational origin in contact geometry.     

Dynamics for a quantum parliament

In this paper, we propose a dynamical approach based on the Gorini–Kossakowski–Sudarshan–Lindblad equation for a problem of decision making. More specifically, we consider what was recently called a quantum parliament, asked to approve or not a certain law, and we propose a model of the connections between the various members of the parliament, proposing in particular some special form of the interactions giving rise to a collaborative or noncollaborative behavior.     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.