A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

Free subgroups of branch groups

A procedure is described for constructing branch groups on the binary tree, which yields in particular finitely generated branch groups with non-cyclic free subgroups.     

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

Norm inequalities of periodic functions and their derivatives

If v is a function on [0,1] such that v^(j) is absolutely continuous and $v^{(j)}(0) + v^{(j)}(1) = 0$ for $0 \leq j \leq n-1$, the inequality in L^p norm between v and the derivative v⁽ⁿ⁾ is proved if $v^{(n)} \in L^{p}[0,1]$, and is shown by examples to be best possible when p = 1, 2 and $\infty$. The method of proof relies on the explicit form of the Green functions of some boundary value problems obtained recently and is applied to refine some related inequalities by J. Favard and D. G. Northcott. Received: 26 April 2002     

Finitely based varieties of pointed-groups

The links between finite basis results for varieties of pointed-groups and those for abstract groups are discussed. It is shown that every variety of nilpotent by abelian-of-finite-exponent pointed-groups is finitely based.Received: 13 October 2000     

Semigroup varieties with a small number of term operations

We give an equational characterization of (varieties of) semigroups having a p_n-sequence bounded above by a polynomial function of n. This is achieved by studying the syntactical connections between certain semigroup identities and their equational consequences.     

Extension and lifting of polynomials

We characterize the Banach spaces enjoying polynomial lifting properties, and the spaces that admit compact extensions of polynomials.Received: 20 June 2002     

When an algebraic frame is regular

It is shown that an algebraic frame L is regular if and only if its compact elements are complemented. More generally, it is shown that each pseudocomplemented element is regular if and only if each , with c compact, is complemented. With a mild assumption on L, each , with c compact, is regular precisely when for any two minimal primes p and q of L. These results are then interpreted in various frames of subobjects of lattice-ordered groups and f-rings.