The algebra of generalized jacobifields

We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of symmetric pseudotensor fields of type (k, 1) and (k+1,1), k≥0, defined on the manifold. We establish an isomorphism between the space of infinitesimal automorphisms of fixed type and the space ℌ_k of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms of the connection. It is shown that the spaces ℌ_k are finite-dimensional and form a graduated Lie algebra ℌ=⊕ _k=0 ^∞ ℌ_k. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1). Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 222–244. Translated by V. S. Kal’nitskii.     

Equivariant framability of involutions on homotopy spheres

Let G be the group Z₂. Denote byR ^n,k theR ^n+k with non trivial G-action on the first n coordinates. Let ^n,k be the trivial bundle with fibreR ^n,k. We say that a G-manifold M is (n,k)-framable if t(M)= =^n,k in KO_G(M) with t(M) the tangent bundle of M. We show that if G acts on a homotopy sphere ^n+k such that the fixed point set is a k-dimensional homotopy sphere then is (n,k)-framable.     

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k−?)-forms for various integers ?. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.     

Obstructions to embeddings of bundles of matrix algebras in a trivial bundle

We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle A _k → X whose fiber is a complex matrix algebra M _k (?) in a trivial bundle with fiber M _kl (?) under the assumption that (k, l) = 1. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group PGL _k (?) of the bundle A _k to SL _k (?), which coincides with the first Chern class c ₁(ξ _k ) reduced modulo k under the assumption that A _k ≌ End(ξ _k ) for some vector ? _k -bundle ξ _k → X. If the first obstruction vanishes, then A _k ≌ End( $\tilde \xi _k $ ) for some vector ? ^k bundle ξ _k → X with structure group SL _k (?), and the second obstruction is c ₂( $\tilde \xi _k $ )modk ∈ H ⁴(X, ?/k?). Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.     

Characterizations of multivariate normality. I. Through independence of some statistics

It is established that a vector variable (X₁, …, X_k) has a multivariate normal distribution if for each X_i the regression on the rest is linear and the conditional distribution about the regression does not depend on the rest of the variables, provided the regression coefficients satisfy some mild conditions. The result is extended to the case where X_i themselves are vector variables.     

Existence and uniqueness of the solution,separation for certain second order elliptic differential equation

In this paper, we study conditions under which Schrodinger type operators with a matrix potential is separated and Schrodinger equation has a unique solution in the weighted space L_2,k(Rⁿ)^l, where l is any natural number and k ε C¹(Rⁿ) is a positive function     

Flow prolongation of some tangent valued forms

We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frölicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.     

Generalized Adjunction of k‐Very Ample Vector Bundles on Algebraic Surfaces II

We study the k‐very ampleness of the adjoint bundle K_S + det E associated to a (k — 1)‐very ample vector bundle E with degree greater than or equal to 4k + 5 on an algebraic surface S. We classify polarized surfaces (S, E) which the k‐very ampleness of K_S + det E fails.     

On the multi‐colored Ramsey numbers of cycles

For a graph L and an integer k≥2, R_k(L) denotes the smallest integer N for which for any edge‐coloring of the complete graph K_N by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdos conjectured that, for an odd cycle C_n on n vertices, They proved the case when k = 2 and also provided an upper bound R_k(C_n)≤(k+ 2)!n. Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k≥4, When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that R_k(C_n)≥(k?1)n?2k+ 4. Here we prove that if n is even, then © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 169‐175, 2012     

On the descent of Levi factors

Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose ? is a finite Galois extension of k for which the extension degree [? : k] is relatively prime to p. If G _/? has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups ${R \simeq Lie(R)}$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If ${G_{{/k}_{sep}}}$ has a Levi decomposition for a separable closure k _sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which ${G_{{/k}_{sep}}}$ has a Levi decomposition, but G itself has no Levi decomposition.     

Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v₁,…,v_m, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ₂-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle. More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.     

A definition of estimator efficiency ink-parameter case

Summary The paper introduces a new definition of efficiency in the multiparameter case (θ₁,...,θ_k) when the variance-covariance matrix of the vector estimator (t ₁, ...t _k) exists. The definition is also applicable to the asymptotically unbiased estimators. The basic idea is that, as we want in general to estimate some function g(θ₁,...θ_k) of the parameters, efficiency of the vector estimator shall be defined as the smallest efficiency of the estimatorg(t ₁, ...t _k),g being regular. It is shown that this definition is asymptotically equivalent to the one obtained by any linear combination of the estimators, as it happens, naturally, for quantile estimation in the location-dispersion case. This efficiency is larger than Cramér efficiency which is, thus, not attained, apart from a very exceptional case. Finally, a lower bound for the asymptotic variance is obtained.     

Heat kernel expansions in vector bundles

Let M be a complete connected smooth (compact) Riemannian manifold of dimension n. Let Π:V→M be a smooth vector bundle over M. Let be a second order differential operator on M, where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M. Let k_t(−,−) be the heat kernel of V relative to L. In this paper we will derive an exact and an asymptotic expansion for k_t(x,y₀) where y₀ is the center of normal coordinates defined on M, x is a point in the normal neighborhood centered at y₀. The leading coefficients of the expansion are then computed at x=y₀ in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M, of the vector bundle V, and of the vector bundle section ? and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean-Singer).     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Generalized Adjunction of k-Very Ample Vector Bundles on Algebraic Surfaces

We study the k-very ampleness of the adjoint bundle K _S+det E associated to a k-very ample vector bundle E on an algebraic surface S. We extend the results of Beltrametti and Sommese to the vector bundle case and give the classification of the pairs (S, E) such that the preservation of k-very ampleness fails.     

Solvability of fourth‐order Sturm‐Liouville problems with abstract linear functionals in boundary‐transmission conditions

This study deals with the solvability of one nonclassical boundary‐value problem for fourth‐order differential equation on two disjoint intervals I₁=(−1,0)and I₂=(0,1). The boundary conditions contain not only endpoints x=−1and x=1but also a point of interaction x=0, finite number internal points x_jki∈I_j and abstract linear functionals S_k. So, our problem is not a pure differential one. We investigate such important properties as isomorphism, Fredholmness and coerciveness with respect to the spectral parameter. Note that the obtained results are new even in the case of the boundary conditions without internal points x_jki and without abstract linear functionals S_k.     

Kalman滤波的一种改型

对于n维状态x∈R~n与不完全观测y∈R~m(m相似文献