On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.     

On the geometry of orthonormal frame bundles

In this paper we are studying the geometry of orthonormal frame bundles over Riemannian manifolds, which are equipped, as submanifolds of the full frame bundles, by the induced Sasaki–Mok metric. All kinds of curvatures are calculated and many geometric results are proved. It seems that the geometry of the orthogonal frame bundles is much more interesting (and more “flexible”) than that of the general frame bundles studied earlier by L. A. Cordero and M. de León. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ? E, then deduce its G × ?₂-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ? _n  × ?₂-graded identities of the algebras M _n (E). Moreover we give the graded cocharacter sequence of M ₂(E), and show that M ₂(E) is PI-equivalent to M _1,1(E) ? E. This fact is a particular case of a more general result obtained by Kemer.     

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R ⁿ . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.     

Entropy and ergodic probability for differentiable dynamical systems and their bundle extensions

We answer a problem of Liao [S.T. Liao, Standard systems of differential equations and obstruction sets—from linearity to perturbations, in: System Researches, Proceedings Dedicated to the 85th Anniversary of Qian Xue-Sen, Zhejiang Education Press, Hangzhou, China, 1996, pp. 279-290 (in Chinese)]: A C¹ vector field or a C¹ diffeomorphism on an n-dimensional manifold has equal entropy with that of its bundle extensions. We also prove that each ergodic probability with simple Lyapunov spectrum has at most n²n! covering probabilities on each bundle extension.     

The Identities and the Central Polynomials of the Infinite Dimensional Unitary Grassmann Algebra Over a Finite Field

We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field.     

Contractions on a manifold polarized by an ample vector bundle

A complex manifold of dimension together with an ample vector bundle on it will be called a generalized polarized variety. The adjoint bundle of the pair is the line bundle . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case . If this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map from onto a normal projective variety with connected fiber and such that , for some ample line bundle on . We describe those contractions for which . We extend this result to the case in which has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which for every fiber and is birational.

     

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C¹-Hamiltonian systems on the cotangent bundle of a C³-smooth compact manifold M without boundary, of a time 1-periodic C²-smooth Hamiltonian H:R×T^*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T^*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T^*M. If M is C⁵-smooth and dimM>1, H is of C⁴ class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.     

The Identities and Central Polynomials of the Finite Dimensional Unitary Grassmann Algebras Over a Finite Field

We describe the T-ideal of identities and the T-space of central polynomials for the unitary finite dimensional Grassmann algebra over a finite field.     

Line bundles for which a projectivized jet bundle is a product

We characterize the triples , consisting of line bundles and on a complex projective manifold , such that for some positive integer , the -th holomorphic jet bundle of , , is isomorphic to a direct sum .

     

Connections and Higgs fields on a principal bundle

Let M be a compact connected Kähler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle ε _G over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (ε _G (L(G)), θ _l ) the Higgs L(G)-bundle associated with (ε _G , θ). The Higgs bundle (ε _G , θ) will be called semistable (respectively, stable) if (ε _G (L(G)), θ _l ) is semistable (respectively, stable). A semistable Higgs G-bundle (ε _G , θ) will be called pseudostable if the adjoint vector bundle ad(ε _G (L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.     

Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

Here we study vector bundles E on the Hirzebruch surface F _e such that their twists by a spanned, but not ample, line bundle M = _Fe (h + ef) have natural cohomology, i.e. h ⁰(F _e , E(tM)) > 0 implies h ¹(F _e , E(tM)) = 0.      

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

熵—证券投资组合风险的一种新的度量方法

本文在研究马科维茨 ( Markowitz)证券投资组合模型的基础上 ,分析了该模型用方差度量风险的缺陷 ,进而提出用熵作为风险的度量方法 ,改进马科维茨 ( Markowitz)证券投资组合模型 ,并建立新的证券投资组合优化模型     

参变极值问题的信息凝聚分布与Boltzmann极大熵函数

该文利用Ｂｏｌｔｚｍａｎｎ 熵概念给出了参变极值问题最优解的一种积分极限表达式和极值函数的极大熵函数,讨论了它们一致收敛性的要求并给出了极大熵函数一致收敛的一个充分条件,将之应用到全局最优解问题得到了全局最优解和最优值的一种显表示,最后还探讨了极大熵函数在一类双层规划问题求解中的应用．     

On the use of the rotation minimizing frame for variational systems with Euclidean symmetry

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel, or Bishop frame. Such systems have previously been studied using the Frenet–Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant calculus for Lie group–based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether's conservation laws as well as the Euler–Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the Euler–Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of proteins, nucleid acids, and polymers.     

Solution bundle for a class of impulsive differential inclusions on Banach spaces

In this paper a class of impulsive differential inclusions is investigated. The existence of solution bundle is proved. And we also construct a nonlinear semigroup of operators on cb(E) (closed-bounded subset of E) to describe the set of attainable states.     

More on a New Concept of Entropy and Information

An alternative notion of entropy called CRE is proposed in [Ra1] Rao et al. (IEEE Trans. Inf. Theory 50, 2004). This preserves many of the properties of Shannon Entropy and possesses mathematical properties, which we hope will be of use in statistical estimates. In this article, we develop some more mathematical properties of CRE, show its relation to the L log L class, and characterize among others the Weibull distribution.