Following Jacobi’s geometrization of Lagrange’s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor (U + h), where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M ^* ≃ S ²(1/2).

In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve (which only records the change of shape) in the case of h h ≠ 0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonicity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.

It is shown that an n × n matrix of continuous linear maps from a pro-C ^*-algebra A to L(H), which verifies the condition of complete positivity, is of the form [V ^* T _ijΦ( · )V] _i,j=1 ⁿ, where Φ is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and [T _ij] _{n i,j=1} ⁿ is a positive element in the C ^*-algebra of all n × n matrices over the commutant of Φ(A) in L(K). This generalizes a result of C. Y. Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709–712. Also, a covariant version of this construction is given.

Let H be a cosemisimple Hopf algebra over a field k, and π: A → H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B = LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.

Let T _2k+1 be the set of trees on 2k+1 vertices with nearly perfect matchings and α(T) be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T _2k+1. Specifically, 10 trees T ₂,T ₃,⋯,T ₁₁ and two classes of trees T(1) and T(12) in T _2k+1 are introduced. It is shown in this paper that for each tree T ₁ ^′, T ₁ ^″ ∈ T(1) and T ₁₂ ^′, T ₁₂ ^″ ∈ T(12) and each i, j with 2 ≤ i < j <-11, α(T ₁ ^′) = α(T ₁ ^″) > α(T _i) > α(T _j) > α(T ₁₂ ^′) = α(T ₁₂ ^″). It is also shown that for each tree T with T ∈ T _2k+1 (T(1) ∪ {T ₂,_{T 3},⋯,T ₁₁} ∪ T(12)), α(T ₁₂ ^′) > α(T).

It is well known that certain isotopy classes of pseudo-Anosov maps on a Riemann surface $\tilde S$ of non-excluded type can be defined through Dehn twists $t_{\tilde \alpha } $ and $t_{\tilde \beta } $ along simple closed geodesics $\tilde \alpha $ and $\tilde \beta $ on $\tilde S$, respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane $\mathbb{H}$ so that ${\mathbb{H}}/G \cong \tilde S$. For any point a ∈ $\tilde S$ define $S = \tilde S\backslash \{ a\} $. In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on $\tilde S$ obtained from Thurston’s construction.

This study focuses on the anisotropic Besov-Lions type spaces B _p,θ ^l(Ω;E ₀,E) associated with Banach spaces E ₀ and E. Under certain conditions, depending on l = (l ₁, l ₂,⋯, l _n) and α = (α₁, α₂, ⋯, α _n), the most regular class of interpolation space E _α between E ₀ and E are found so that the mixed differential operators D ^α are bounded and compact from B _p,θ ^l+s(Ω;E ₀,E) to B _p,θ ^s(Ω;E _α). These results are applied to concrete vector-valued function spaces and to anisotropic differential-operator equations with parameters to obtain conditions that guarantee the uniform B separability with respect to these parameters. By these results the maximal B-regularity for parabolic Cauchy problem is obtained. These results are also applied to infinite systems of the quasi-elliptic partial differential equations and parabolic Cauchy problems with parameters to obtain sufficient conditions that ensure the same properties.