Nonexistence of maximizers for the functional of the centroaffine Minkowski problem

The centroaffine Minkowski problem is studied, which is the critical case of the L _p -Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions. In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.     

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E₁, F₁) ? · · · ? (Ek; Fk); where F _j := F |E _j , and each (E _j , F _j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).     

Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = Δ_E² + q to be separated in the space L² (M) on a complete Riemannian manifold (M,g) with metric g, where Δ_E is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L² (M) if for all u ∈ L² (M) such that Au ∈ L² (M) we have Δ_E²u ∈ L² (M) and qu ∈ L² (M).     

Traces of Quantized Canonical Transformations Localized on a Finite Set of Points

For an embedding i : X ? M of smooth manifolds and a Fourier integral operator Φ on M defined as the quantization of a canonical transformation g: T*M \ {0} → T*M \ {0}, we consider the operator i*Φi* on the submanifold X, where i* and i_* are the boundary and coboundary operators corresponding to the embedding i. We present conditions on the transformation g under which such an operator has the form of a Fourier integral operator associated with the fiber of the cotangent bundle over a point. We obtain an explicit formula for calculating the amplitude of this operator in local coordinates.     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

Kreĭn space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H²(D²). A closed subspace M in H²(D²) is called a submodule if z _i M ? M (i = 1, 2). An associated integral operator (defect operator) C _M captures much information about M. Using a Kre?n space indefinite metric on the range of C _M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

SCAP-subalgebras of Lie algebras

A subalgebra H of a finite dimensional Lie algebra L is said to be a SCAP-subalgebra if there is a chief series 0 = L₀ ? L₁ ?... ? L_t = L of L such that for every i = 1, 2,..., t, we have H + L_i = H + L_i-1 or H ∩ L_i = H ∩ L_i-1. This is analogous to the concept of SCAP-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its SCAP-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.     

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.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian \(L:T^{(k)}Q\rightarrow {\mathbb {R}}\) with \(k\ge 1\), the resulting discrete equations define a generally implicit numerical integrator algorithm on \(T^{(k-1)}Q\times T^{(k-1)}Q\) that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian \(L_d^e\) using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of \(L_d^e\), we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.     

Coloring number and on-line Ramsey theory for graphs and hypergraphs

Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G ₀=(V,E ₀), where E ₀=Ø and V is determined by Builder. On the ith round Builder constructs a new edge e _i (distinct from previous edges) and sets G _i =(V,E _i ), where E _i =E _i?1∪{e _i }. Painter responds by coloring e _i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K _s ^t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges.We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K _s ^t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H ₀ = (V,Ø) with an arbitrary finite number of vertices and no edges. Let H _i?1=(V _i?1,E _i?1) be the hypergraph constructed in the first i ? 1 rounds. On the i-th round Presenter plays by presenting a p-subset P _i ?V _i?1 and Chooser responds by choosing an s-subset X _i ?P _i . The vertices in P _i ? X _i are discarded and the edge X _i added to E _i?1 to form E _i . Presenter wins the survival game if H _i contains a copy of K _s ^t for some i. We show that for positive integers p,s,t with s≤p, Presenter has a winning strategy.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E _n , Euler polynomials E _n (x), generalized Euler numbers E _n (a, b), generalized Euler polynomials E _n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E _n , E _n (x), E _n (a, b), E _n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).