Hierarchical ferromagnetic vector spin model possessing the Lee-Yang property. Thermodynamic limit at the critical point and above

The hierarchical ferromagneticN-dimensional vector spin model as a sequence of probability measures onR ^N is considered. The starting element of this sequence is chosen to belong to the Lee-Yang class of measures that is defined in the paper and includes most known examples (⁴ measures, Gaussian measures, and so on). For this model, we probe two thermodynamic limit theorems. One of them is just the classical central limit theorem for weakly dependent random vectors. It describes the convergence of classically normed sums of spins when temperature is sufficiently high. The other theorem describes the convergence of more than normally normed sums that holds for some fixed temperature. It corresponds to the strong dependence of spins, which appears at the critical point of the model.     

Flat Complex Vector Bundles,the Beltrami Differential and W-Algebras

Since the appearance of the paper by Bilal et al. in 1991, it has been widely assumed that W-algebras originating from the Hamiltonian reduction of an SL(n,C)-bundle over a Riemann surface give rise to a flat connection, in which the Beltrami differential may be identified. In this Letter, it is shown that the use of the Beltrami parametrization of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows the construction of the above mentioned flat connection. It is stressed that the modulus of the Beltrami differential is of necessity less than one, and that solutions of the so-called Beltrami equation give rise to an orientation-preserving smooth change of local complex coordinates. In particular, the latter yields a smooth equivalence between flat complex vector bundles. The role of smooth diffeomorphisms which induce equivalent complex structures is specially emphasized. Furthermore, it is shown that, while the construction given here applies to the special case of the Virasoro algebra, the extension to flat complex vector bundles of arbitrary rank does not provide generalizations of the Beltrami differential usually considered as central objects for such non-linear symmetries.     

Classification of semispaces according to their types in infinite-dimensional vector spaces

It is shown that each semispaceC X naturally generates a relation of complete preorder onX with respect to which the pair (X C, C) is a cut ofX. By identifying the type of the semispace with the type of the cut generated by this semispace, the semispaces are classified according to their types. The obtained classification extends the classification of semispaces in finite-dimensional vector spaces due to Martinez-Legaz and Singer to infinite-dimensional spaces.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 191–198, August, 1998.     

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side was carried over by
Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of is a necessary condition here if is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in by taking coordinate functionals).

     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Let be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H _Z to Z is very ample. Triplets are classified under the assumption that (Z,H _Z ) has a smooth bielliptic curve section of genus ≥ 3 with .      

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).