Finite -arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite -arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of -transitive Cayley graphs with such that all -transitive Cayley graphs of valency are their normal covers. This indicates that -arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .

     

On restricted weak type : The continuous case

Let denote the unit circle. An example of a sublinear translation-invariant operator acting on is given such that is of restricted weak type but not of weak type .

     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

The vanishing of implies that is regular

Let be an excellent local ring of positive prime characteristic. We show that if , then is regular. This improves a result of Schoutens, in which the additional hypothesis that was an isolated singularity was required for the proof.

     

Convolution operators induced by approximate identities and pointwise convergence in spaces

Given a sequence of kernels for which the operators converge a.e. in all spaces, , a perturbation method is provided with the property that the modified convolution operators converge pointwise only in selective spaces.

     

Algebraic functions with even monodromy

Let be a compact Riemann surface of genus and be an integer. We show that admits meromorphic functions with monodromy group equal to the alternating group

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

The content of a Gaussian polynomial is invertible

Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.

     

Filling analytic sets by the derivatives of -smooth bumps

If is an infinite-dimensional Banach space, with separable dual, and is an analytic set such that any point can be reached from  by a continuous path contained (except for the point ) in the interior of , then is the range of the derivative of a -smooth function on  with bounded nonempty support.