Vertical Subspaces Research Articles

On convergence of intrinsic volumes of Riemannian manifolds

Let \(\pi :M\rightarrow B\) be a Riemannian submersion of two compact smooth Riemannian manifolds, B is connected. Let \(M(\varepsilon )\) denote the manifold M equipped with the new Riemannian metric which is obtained from the original one by multiplying by \(\varepsilon \) along the vertical subspaces (i.e. along the fibers) and keeping unchanged along the (orthogonal to them) horizontal subspaces. Let \(V_i(M(\varepsilon ))\) denote the ith intrinsic volume. The main result of this note says that \(\lim _{\varepsilon \rightarrow +0}V_i(M(\varepsilon ))=\chi (Z) V_i(B)\) where \(\chi (Z)\) denotes the Euler characteristic of a fiber of \(\pi \).

Продолжения аффинной связности и горизон­тальных векторов

The linear frame bundle over a smooth manifold is considered. The mapping dе defined by the differentials of the first-order frame e is a lift to the normal N, i. e., a space complementing the first-order tangent space to the second-order tangent space to this bundle. In particular, the map­ping defined by the differentials of the vertical vector of this frame is a vertical lift into normal N. The lift dе allows us to construct a prolongation both of the tangent space and its vertical subspace into the second-order tangent space, more precisely into the normal N. The normal lift dе defines the normal prolon­gation of the tangent space (i. e. the normal N) and its vertical subspace. The vertical lift defines the vertical prolongation of the tangent space and its vertical subspace. The differential of an arbitrary vector field on the linear frame bundle is a complete lift from the first-order tangent space to the second-order tangent space to this bundle. It is known that the action of vector fields as differential operators on functions coincides with action of the differentials of these functions as 1-forms on these vector fields. Horizontal vectors played a dual role in the fibre bundle. On the one hand, the basic horizontal vectors serve as opera­tors for the covariant differentiation of geometric objects in the bundle. On the other hand, the differentials of these geometric objects can be con­sidered as forms (including tangential-valued ones) and their values on basic horizontal vectors give covariant derivatives of these geometric ob­jects. For objects which covariant derivatives require the second-order con­nection, the covariant derivatives are equal to the values of the differen­tials of these objects on horizontal vectors in prolonged affine connectivi­ty. Prolongations of the basic horizontal vectors, i. e., the second-order horizontal vectors for prolonged connection, were constructed. The sec­ond-order tangent space is represented as a straight sum of the first-order tangent space, vertical prolongations of the vertical and horizontal sub­spaces, and horizontal prolongation of the horizontal subspace.

Diagnosis of Breast Cancer Tissues Using 785 nm Miniature Raman Spectrometer and Pattern Regression.

For achieving the development of a portable, low-cost and in vivo cancer diagnosis instrument, a laser 785 nm miniature Raman spectrometer was used to acquire the Raman spectra for breast cancer detection in this paper. However, because of the low spectral signal-to-noise ratio, it is difficult to achieve high discrimination accuracy by using the miniature Raman spectrometer. Therefore, a pattern recognition method of the adaptive net analyte signal (NAS) weight k-local hyperplane (ANWKH) is proposed to increase the classification accuracy. ANWKH is an extension and improvement of K-local hyperplane distance nearest-neighbor (HKNN), and combines the advantages of the adaptive weight k-local hyperplane (AWKH) and the net analyte signal (NAS). In this algorithm, NAS was first used to eliminate the influence caused by other non-target factors. Then, the distance between the test set samples and hyperplane was calculated with consideration of the feature weights. The HKNN only works well for small values of the nearest-neighbor. However, the accuracy decreases with increasing values of the nearest-neighbor. The method presented in this paper can resolve the basic shortcoming by using the feature weights. The original spectra are projected into the vertical subspace without the objective factors. NAS was employed to obtain the spectra without irrelevant information. NAS can improve the classification accuracy, sensitivity, and specificity of breast cancer early diagnosis. Experimental results of Raman spectra detection in vitro of breast tissues showed that the proposed algorithm can obtain high classification accuracy, sensitivity, and specificity. This paper demonstrates that the ANWKH algorithm is feasible for early clinical diagnosis of breast cancer in the future.

Sparse Frequency Waveform Design for Radar-Embedded Communication

According to the Tag application with function of covert communication, a method for sparse frequency waveform design based on radar-embedded communication is proposed. Firstly, sparse frequency waveforms are designed based on power spectral density fitting and quasi-Newton method. Secondly, the eigenvalue decomposition of the sparse frequency waveform sequence is used to get the dominant space. Finally the communication waveforms are designed through the projection of orthogonal pseudorandom vectors in the vertical subspace. Compared with the linear frequency modulation waveform, the sparse frequency waveform can further improve the bandwidth occupation of communication signals, thus achieving higher communication rate. A certain correlation exists between the reciprocally orthogonal communication signals samples and the sparse frequency waveform, which guarantees the low SER (signal error rate) and LPI (low probability of intercept). The simulation results verify the effectiveness of this method.

Bochner flatness of tangent bundles with g-natural almost Hermitian metrics

We consider g-natural metrics on the tangent bundle of a Riemannian manifold together with the almost complex structure which reverses the horizontal and vertical subspaces. This narrows the class of g-natural metrics to metrics conformally equivalent to the Sasaki metric on the tangent bundle with a restriction on the conformal factor. We then show that such a g-natural almost Hermitian structure is Bochner flat if and only if it is conformally equivalent to the Sasaki metric when the base manifold is flat and with the same restriction on the conformal factor.

An Affine Scaling Interior Trust-Region Algorithm Combining Backtracking Line Search with Filter Technique for Nonlinear Constrained Optimization

In this article, an affine scaling interior trust-region algorithm which employs backtracking line search with filter technique is presented for solving nonlinear equality constrained programming with nonnegative constraints on variables. At current iteration, the general full affine scaling trust-region subproblem is decomposed into a pair of trust-region subproblems in vertical and horizontal subspaces, respectively. The trial step is given by the solutions of the pair of trust-region subproblems. Then, the step size is decided by backtracking line search together with filter technique. This is different from traditional trust-region methods and has the advantage of decreasing the number of times that a trust-region subproblem must be resolved in order to determine a new iteration point. Meanwhile, using filter technique instead of merit function to determine a new iteration point can avoid the difficult decisions regarding the choice of penalty parameters. Under some reasonable assumptions, the new method possesses the property of global convergence to the first-order critical point. Preliminary numerical results show the effectiveness of the proposed algorithm.

The geometry of the tangent bundle and the relativistic kinetic theory of gases

This paper discusses the relativistic kinetic theory for a simple collisionless gas from a geometric perspective. We start by reviewing the rich geometrical structure of the tangent bundle TM of a given spacetime manifold, including the splitting of the tangent spaces of TM into horizontal and vertical subspaces and the natural metric and symplectic structure it induces on TM. Based on these structures we introduce the Liouville vector field L and a suitable Hamiltonian function H on TM. The Liouville vector field turns out to be the Hamiltonian vector field associated to H. On the other hand, H also defines the mass shells as Lorentzian submanifolds of the tangent bundle. When restricted to these mass shells, the projection of the integral curves of L on the base manifold describes a family of future directed timelike geodesics. A simple collisionless gas is described by a distribution function on a particular mass shell, satisfying the Liouville equation. Together with the Liouville vector field the distribution function can be thought of as defining a fictitious incompressible fluid on the mass shells, with associated conserved current density. Flux integrals of this current density provide the averaged properties of the gas, while suitable fiber integrals of the distribution function define divergence-free tensor fields on the spacetime manifold such as the current density and stress–energy tensor. Finally, we discuss the relationship between symmetries of the spacetime manifold and symmetries of the distribution function. Taking advantage of the natural metric and symplectic structure on TM, we show that groups of isometries G of the spacetime manifold lift naturally to groups of isometries and symplectic flows on the tangent bundle. Motivated by these properties, we define a distribution function to be G-invariant whenever it is invariant under the lifted isometries. As a first application of our formalism we derive the most general spherically symmetric distribution function on any spherically symmetric spacetime and write the Einstein–Liouville equations as effective field equations on the two-dimensional radial manifold. As a second application we derive the most general collisionless distribution function on a Kerr black hole spacetime background.

Topological Shift Spaces

Abstract Two of the problems listed in [14, 74.17] ask to prove or disprove the following statements: A) For each differentiable planar map ƒ : IR2 → IR2 the set of all differentials defines a spread of IR4. B) If the differentials of a differentiable map ƒ : IR2 → IR2 define a spread of IR4 then the map ƒ is planar. By restricting to vertical 3-dimensional subspaces, we get the notion of a 3-dimensional shift space, and for differentiable shift spaces we may formulate analogous problems A′ and B′. Under the additional assumption that there exists a 1-dimensional group of shears, we prove A′ and B′ for 3-dimensional shift spaces and—as a corollary—also A and B for 4-dimensional shift planes.

Interior point projected reduced Hessian method with trust region strategy for nonlinear constrained optimization

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems, a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.

An affine scaling projective reduced Hessian algorithm for minimum optimization with nonlinear equality and linear inequality constraints

In this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by QR decomposition of an affine scaling matrix and an orthonormal basis on the null subspace. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. Combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. The second-order correction step and a nonmonotonic criterion are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases, respectively.

An affine scaling interior point backtracking algorithm for nonlinear constrained optimisation

AbstractIn this paper we propose a new affine scaling interior trust region algorithm with a nonmonotonic backtracking technique for nonlinear equality constrained optimisation with nonnegative constraints on the variables. In order to deal with large problems, the general full trust region subproblem is decomposed into a pair of trust region subproblems in horizontal and vertical subspaces. The horizontal trust region subproblem in the algorithm is defined by minimising a quadratic function subject only to an ellipsoidal constraint in a null tangential subspace and the vertical trust region subproblem is defined by the least squares subproblem subject only to an ellipsoidal constraint. By adopting Fletcher's penalty function as the merit function, combining a trust region strategy and a nonmonotone line search, the mixing technique will switch to a backtracking step generated by the two trust region subproblems to obtain an acceptable step. The global convergence of the proposed algorithm is proved while maintaining a fast local superlinear convergence rate, which is established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some highly nonlinear cases.

An interior affine scaling projective algorithm for nonlinear equality and linear inequality constrained optimization

In this paper, we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. By introducing the Fletcher's penalty function as the merit function, trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. The global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases.

Lower bounds on volumes of vector fields

Let M be an n-dimensional closed (compact without boundary) Riemannian manifold, and let X be a unit smooth vector field on M. It can be looked at as a n-dimensional submanifold in the unit tangent bundle U (M). The latter can be endowed by the canonical metric (introduced by Sasaki), if we decompose every tangent space to U (M) into horizontal and vertical subspaces and pull back the initial metric of M to each of them (cf. [2]). Following Gluck and Ziller [3] we call the n-dimensional volume of this submanifold the volume of the vector field X. If VX denotes the covariant derivative tensor (so VX(x) is a linear operator in TxM for x ~ M) then it is shown in [3] that

The geodesic spray, the vertical projection, and Raychaudhuri's equation

The metric connection on a space-time manifoldM defines on its tangent bundleTM a distribution of subspaces complementary to the vertical subspaces and therefore called horizontal. We give a formula for the Lie derivative with respect to the geodesic spray of the tensor field onTM which defines projection onto the vertical subspace along the horizontal subspace; and we show that this formula is a universal version of the equation, for a geodesic local vector field onM, whose trace is Raychaudhuri's equation.

Graded gauge theory

The mathematical background for a graded extension of gauge theories is investigated. After discussing the general properties of graded Lie algebras and what may serve as a model for a graded Lie group, the graded fiber bundle is constructed. Its basis manifold is supposed to be the so-called superspace, i.e. the product of the Minkowskian space-time with the Grassmann algebra spanned by the anticommuting Lorentz spinors; the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the SU(N) Lie algebra. The connection and curvature are defined then on this bundle; the two different gradings are either independent of each other, or may be unified in one common grading, which is equivalent to the choice of the spin-statistics dependence. The Yang-Mills lagrangian is investigated in the simplified case. The conformal symmetry breaking is discussed, as well as some other physical consequences of the model.

HORIZONTAL DISTRIBUTIONS ON SPACES OF FIBERS

Abstract Geometric methods for systems of partial differential equations and multiple integral problems in the calculus of variations lead naturally to differentiable manifolds that resemble fiber bundles but do not possess a structure group; in terms of local coordinates, π:B→Mn|(xi, qα)→(xi), dim(B) = N + n, dim(Mn) = n. The standard notions of horizontal distributions, horizontal and vertical subspaces of T(B), T(B) = V(B) ⊕ H(B), horizontal lifts of curves in Mn, and horizontal and vertical dual subspaces with Λ1(B) = V*(B) ⊕ H*(B) are shown to be well defined in B. The absence of a structure group is compensated for by an analysis based on the homogeneous ideals V and H that are generated by the canonical bases of V*(B) and H*(B), respectively. The differential system constructed from the generators of the horizontal ideal is shown to lead to a unique system of connection 1-forms and torsion 2-forms under the requirements that they have vacuous intersections with the horizontal ideal. The horizontal ...

Covariant objects and invariant equations on fiber bundles

Let P(M4,G) be a principal fiber bundle over the Minkowskian space–time M4 with the structural group G. The group G is supposed to be a compact and semisimple Lie group. Let A be a connection form on P(M4,G) and F=DA its curvature form. Let gG be the Cartan–Killing metric on G, and gM4 the Minkowskian metric on M4. Let us define dπ : TP→TM4, the differential of the canonical projection from P onto M4. Then we can define a scalar product for any two vectors from P(M4,G): gP(X,Y)=gG(A(X),A(Y))+gM4Q (dπ(X),dπ(Y)). In this metric the horizontal and vertical subspaces of the connection A are orthogonal to each other. Next, we construct the Clifford algebra corresponding to the metric gP. The metric gP can be always diagonalized locally to give diag((3+N)+,1−), where N is the dimension of G. The lowest faithful representation of this algebra, which we call C(3+N,1) is of the dimension K=2[(N+5)/2]. This K-dimensional vector space is called the space of spinors over P(M4,G). We study the decomposition of these spinors into multiplets of Lorentz spinors. We also define the generalized Dirac equation for such a spinor, construct an explicit representation in the case of G=SU(2), and give the formulae for the mass splitting. Finally, the invariant interaction with vector fields over P(M4,G) and scalar multiplets is discussed, together with the physical implications of the coupled equations.