Generic analytical embedding methods for nonstationary systems based on control theory

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weil's main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].

The global feature of the spontaneously broken gauge theory is discussed in the context of fiber bundle. With the example of 'tHooft-Polyakov monopole, we show how the Higgs mechanism makes a nontrivial fiber bundle. The general relation of the topology of Higgs field ancl the structure of fiber bundle is presented, and extension of the argument to a general gauge theory with Higgs mechanism is also macle.

A fibre bundle is a bundle with an additional structure derived from the action of a topological group on the fibres. In the next chapter the notions of fibre bundle and vector bundle are related. As with vector bundles, a fibre bundle has so much structure that there is a homotopy classification theorem for fibre bundles.KeywordsVector BundleOpen NeighborhoodFibre BundleTopological GroupOpen CoveringThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In many areas of theoretical physics we encounter fields defined on a spacetime M taking values in some other space F. For instance, presume F as a real finite-dimensional vector space, then ∅ : M → F is a vector field. More generally, we can consider a family of spaces {Fx}xϵM varying over the points on M, that is ∅(x) ϵ Fx for each x ϵ M. A field ∅ is then understood as a section from the spacetime manifold into the bundle of spaces over M. This is exactly the idea encoded in the mathematical theory of fiber bundles. Namely, fiber bundles provide a tool to describe the global structure of physical fields.

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is SU(3)/U(1)2. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.

We show that a proper algebraic n n -dimensional scheme Y Y admits non-trivial vector bundles of rank n n , even if Y Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional locus in only finitely many points. Moreover, there are such vector bundles with arbitrarily large top Chern number. Applying this to toric varieties, we infer that every proper toric threefold admits such vector bundles of rank three. Furthermore, we describe a class of higher-dimensional toric varieties for which the result applies, in terms of convexity properties around rays.

Poincare bundles and Picard bundles on the Jacobian of a smooth curve have been studied extensively classically. Peter Newstead and his collaborators have made major contributions to the theory of Poincaré and Picard bundles over the moduli spaces of vector bundles of higher ranks. We present old and new results on the Poincare bundles and Picard bundles on the moduli spaces of vector bundles on smooth curves and their generalisations to nodal curves. Finally we consider parabolic vector bundles. We prove the parabolic stability of Poincaré bundles over the moduli spaces of parabolic vector bundles, with parabolic structure of any type, on a smooth curve.

On the commuting variety of a reductive Lie algebra and other related varieties

Deformations of the generalised Picard bundle

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.

In heterotic string theories consistency requires the introduction of a non-trivial vector bundle. This bundle breaks the original ten-dimensional gauge groups E8 × E8 or SO(32) for the supersymmetric heterotic string theories and SO(16) × SO(16) for the non-supersymmetric tachyon-free theory to smaller subgroups. A vast number of MSSM-like models have been constructed up to now, most of which describe the vector bundle as a sum of line bundles. However, there are several different ways of describing these line bundles and their embedding in the ten-dimensional gauge group. We recall and extend these different descriptions and explain how they can be translated into each other.

In recent years there has been some interest in rank2 algebraic (or holomorphic) vector bundles over the complex projective space pn. One of the main reasons for this interest is the connection of such bundles with projective varieties of codimension 2 and in particular with the question whether such varieties are necessarily complete intersections. In [2] Hartshorne formulates the conjecture that every rank2 algebraic bundle over pn splits as the sum of 2 line bundles at least if n > 7. Rank 2 bundles over pa are classified topologically by their Chern classes cl, c 2. In [1] Atiyah and Rees carry out the topological classification for p3 and show the existence of "sufficiently many" rank 2 topological bundles over p4. On p2 and p3 every topological 2-plane bundle has an algebraic structure. It is not known whether this is true for n > 4. In [6] and [-7] non-trivial 2-plane bundles are constructed over P", n>5 , with c a = c 2 = 0 . These are possible candidates for topological bundles without algebraic structure. In this paper we carry the classification of topological 2-plane bundles over P~ a few steps further. Although this by no means solves the classification problem for algebraic bundles, we hope the results may be of some use to those interested in algebraic bundles and projective varieties. In the following we shall not distinguish between a map f:X~BU(2), its homotopy:class and the induced isomorphism class of rank 2 bundles. Thus we shall often write r/$X or q :X~BU(2) . As H*(P";2~) is a truncated polynomial algebra on a 2-dimensional generator x, the Chern classes cl, c 2 are integer multiples of x resp. x 2. Therefore we shall regard them as integers. Let us write

Effect of CF/GF fibre hybrid on impact properties of multi-axial warp knitted fabric composite materials

In this study, thermal properties of plain-woven laminates are analyzed micro-, meso-and macroscopically based on a multiscale approach. For this, the effects of thermal expansion of constituents are incorporated into the micro/meso/macro homogenization method for plain-woven laminates developed by our research group. This method enables us to analyze thermal properties of not only fiber bundles in laminates but also fibers and matrix materials in fiber bundles, in addition to their macroscopic thermal properties. The present method is then applied to the thermal property analysis of plain-woven carbon fiber/epoxy laminates subjected to a macroscopic temperature change from 180°C to 20°C. Two types of carbon fibers, i.e. HTA and P75, are considered in the analysis. It is shown that quite high thermal residual stress can occur in fiber bundles, fibers and a matrix. It is also shown that the present method can predict the change of thermal properties of the laminates depending on the difference of fibers.