A note on symmetric orderings

Let A^n be the completion by the degree of a differential operator of the n-th Weyl algebra with generators x1,…,xn,∂1,…,∂n. Consider n elements X1,…,Xn in A^n of the formXi=xi+∑K=1∞∑l=1n∑j=1nxlpijK−1,l(∂)∂j,where pijK−1,l(∂) is a degree (K−1) homogeneous polynomial in ∂1,…,∂n, antisymmetric in subscripts i,j. Then for any natural k and any function i:{1,…,k}→{1,…,n} we prove∑σ∈Σ(k)Xiσ(1)⋯Xiσ(k)▹1=k!xi1⋯xik,where Σ(k) is the symmetric group on k letters and ▹ denotes the Fock action of the A^n on the space of (commutative) polynomials.

On the iterated normed duals

Several properties of the normed Hom-functor (dual) D and its iterations Dn are exhibited. For instance, D turns every canonical embedding (into the second dual space) to a retraction (of the third dual onto the first one) having for the right inverse the appropriate canonical embedding (of the first dual space into the third one). Some consequences to the direct-sum presentations and quotients of higher dual spaces are considered.

A note on Banach fixed point property

In this short note we consider a sort of converse of the Banach fixed point theorem and prove that a metric space X is complete if and only if, for each closed subspace Y ⊆ X, any contraction f : Y → Y has a fixed point y ∈ Y.

Unitary (g, K) modules of SU(2, 1)

Let G=SU(2,1). In this paper we parametrize irreducible unitary (g,K) modules of G. The parametrization is done in two steps. Firstly, we parametrize irreducible (g,K) modules (Theorem 10). In the second step we find unitary (g,K) modules (Theorem 18). One can compare our results with [4] and [5].

Rational Diophantine sextuples containing two regular quadruples and one regular quintuple

A set of m distinct nonzero rationals {a1,a2,…,am} such that aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m, is called a rational Diophantine m-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

Complete forcing numbers of rectangular polynominoes

Let G be a graph with edge set E(G) that admits a perfect matching M. A forcing set of M is a subset of M contained in no other perfect matchings of G. A complete forcing set of G, recently introduced by Xu et al. [Complete forcing numbers of catacondensed hexagonal systems, J. Combin. Optim. 29(4) (2015) 803-814], is a subset of E(G) on which the restriction of any perfect matching M is a forcing set of M. The minimum possible cardinality of complete forcing sets of G is the complete forcing number of G. In this article, we discuss the complete forcing number of rectangular polyominoes (or grids), i.e., the Cartesian product of two paths of various lengths, and show explicit formulae for the complete forcing numbers of rectangular polyominoes in terms of the lengths.

On classification of morphisms by box-homotopy

In [1] the authors proposed a generalization of the notion of homotopy, a relation called to be box-homotopic, proven to be an equivalence relation on Top(X,Y) and well-adjusted with the composition. In this article we prove that all the mappings of Top(X,Y) are box-homotopic, that is, the classification of morphisms by the box-homotopy relation is the coarsest.

A Note on Curves

There are several rather different approaches to the notion of a curve. We have found the way which assures that, beside an arc, each circle carries the unique curve. As a consequence, each 1-parametrizable set yields at most finitely many curves. Further, all essential properties and well known invariants of a curve are preserved.

L∞ - derivations and the argument shift method for deformation quantization algebras

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an L∞-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.