On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

Two remarks on remotality

We prove that there exists a weakly closed and bounded subset $E$ of $c_0$ which is not remotal from 0, and such that $\overline{\mathrm{co}}\left(E\right)$ is remotal from 0. This answers a question of M. Martín and T.S.S.R.K. Rao. We also present a simple proof of the fact that in every non-reflexive Banach space there exists a closed convex bounded set which is not remotal.     

Two remarks on the gradient method

We display connections between Euler summability, Taylor series, and the gradient method for inversion of linear, symmetric, positive-definite operators on Hilbert space.     

Some properties of basis invariants of the symmetry groupsG(m,p,n), B n m,D n m

We study the properties of the basis invariants of the symmetry groups of the complex polytope and the generalized n-cube γ _n ^m , as well as its subgroups D _n ^m . We give an explicit construction of all the basis invariants of even degree of these groups. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Two remarks on equivelar manifolds

It has been pointed out that a construction for equivelar polyhedral manifolds described in an earlier paper by the present authors may possibly not be performed; an alternative construction is given here. Further, it is shown that certain equivelar polyhedra cannot have equiangular (and hence regular) faces.     

Two remarks on Monge-Ampere equations

Summary We consider real Monge-Ampère equations and we present two new properties of these equations. First, we show the existence of the «first eigenvalue of Monge-Ampère equation» i.e. we show the existence of a positive constant possessing all the properties of the first eigenvalue of a 2-nd order elliptic operator (positivity, uniqueness of the eigenfunction, maximum principle, bifurcation...).The second property concerns variational characterisations of solutions. Both properties are closely related to similar properties of the general class of Hamilton-Jacobi-Bellman equations.     

Two remarks on frequent hypercyclicity

We show that if T:X→X$T:X\to X$ is a continuous linear operator on an F$F$-space X≠{0}$X\ne \left\{0\right\}$, then the set of frequently hypercyclic vectors of T$T$ is of first category in X$X$, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→X$T:X\to X$ is a bounded linear operator on a Banach space X≠{0}$X\ne \left\{0\right\}$ and if T$T$ is frequently hypercyclic (or, more generally, syndetically transitive), then the T^∗$T^{\ast }$-orbit of every non-zero element of X^∗$X^{\ast }$ is bounded away from 0, and in particular T^∗$T^{\ast }$ is not hypercyclic.     

Two remarks on generalized hexagons

In this paper, the point-distance-2-regular hexagons are characterized as the only hexagons for which the intersection sets have size one and containing on ovoidal subspace all the points of which are 3-regular. We also give a characterization of the finite split Cayley hexagon of even order.     

Two remarks on Fourier-Stieltjes transforms

Special refinements of a theorem of Rajchman are given, along with some comments on approximation of certain Fourier-Stieltjes transforms. Work supported in part by the NSF.     

Two remarks on circular arc graphs

In our first remark we observe a property of circular arcs which is similar to the Helly property and is helpful in describing all maximal cliques in circular arc graphs (as well as allowing us to genralize a result of Tucker). Our main result is a new simple characterization of circular arc graphs of clique covering number two. These graphs play a crucial role in recognition algorithms for circular arc graphs, and have been characterized by several authors. Specifically, we show that a graph with clique covering number two is a circular arc graph if and only if its edges can be coloured by two colours so that no induced four-cycle contains two opposite edges of the same colour. Our proof of the characterization depends on the lexicographic method we have recently introduced. Both remarks could be useful in designing efficient algorithms for (maximum cliques in, respectively recognition of) circular arc graphs     

Two remarks on sextic double solids

We study the potential density of rational points on double solids ramified along singular reduced sextic surfaces. Also, we investigate elliptic fibration structures on nonsingular sextic double solids defined over a perfect field of characteristic 5.     

Two remarks on witten'squantum enveloping algebra

In this note we describe two simple tricks derived from work of M. Artin , J. Tate and M. Van den Bergh [2] and apply them to the quantum enveloping algebra discovered by E. Witten [ll] encoding duality in three dimensional Chern-Simons gauge theory with structure group SU(2). In particular, we give a more conceptual interpretation of the dichotony in the representation theory and determine all finite dimensional simple representations.     

Two remarks on the topology of projective surfaces

We will prove two results about the topology of complex projective surfaces. The first result says that if the Shafarevich Conjecture has an affirmative answer in dimension two then the second homotopy group of a smooth projective surface is a torsion-free abelian group. The second result is that for any 2-dimensional function field K/C there is a normal projective simply-connected surface with function field K.