Isomorphic pairs of homogeneous functions and their morphisms

Summary. In this paper we determine all iseomorphic pairs (isomorphic pairs with monotonic, thus continuous isomorphisms) of continuous, strictly increasing, linearly homogeneous functions defined on cartesian squares I ² and J ² of intervals of positive numbers or on their restrictions or and or We prove that, if the iseomorphy is nontrivial, then each homogeneous function is a (weighted) geometric or power mean or a joint pair of such means. In functional equations terminology this means that all nontrivial continuous strictly increasing linearly homogeneous solutions G, H (with the continuous strictly monotonic F also unknown) of the equation on D _< or D _> are weighted geometric or power means, while on I ² they are joint pairs of weighted geometric means or of weighted power means.     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

Presentations of the unit group of an order in a non-split quaternion algebra

We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of . As a consequence a presentation is discovered for the orthogonal group . These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.     

Holomorphic functions and quasiconformal mappings with smooth moduli

We discuss the implication , where f is a holomorphic function (resp., a quasiconformal mapping) on a domain (resp., ) and Λ_ω(G) is the Lipschitz space associated with a majorant ω.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

A family of invariants of rooted forests

Let A be a commutative k-algebra over a field of k and Ξ a linear operator defined on A. We define a family of A-valued invariants Ψ for finite rooted forests by a recurrent algorithm using the operator Ξ and show that the invariant Ψ distinguishes rooted forests if (and only if) it distinguishes rooted trees T, and if (and only if) it is finer than the quantity α(T)=|Aut(T)| of rooted trees T. We also consider the generating function with , where is the set of rooted trees with n vertices. We show that the generating function U(q) satisfies the equation . Consequently, we get a recurrent formula for U_n (n?1), namely, U₁=Ξ(1) and U_n=ΞS_n−1(U₁,U₂,…,U_n−1) for any n?2, where are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well-known quasi-symmetric function invariants of rooted forests are in the family of invariants Ψ and derive some consequences about these well-known invariants from our general results on Ψ. Finally, we generalize the invariant Ψ to labeled planar forests and discuss its certain relations with the Hopf algebra in Foissy (Bull. Sci. Math. 126 (2002) 193) spanned by labeled planar forests.     

The t#-property for integral domains

Generalizing work of Gilmer and Heinzer, we define a t#-domain to be a domain R in which for any two distinct subsets and of the set of maximal t-ideals of R. We provide characterizations of these domains, and we show that polynomial rings over t#-domains are again t#-domains. Finally, we study overrings of t#-domains.     

Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.