A supplement to ‘A note on invariant measure semigroups’

A unified definition of measure semigroup is given and it is shown how the types of invariance for the measure are formulated and what effect they have on the embeddability of an Abelian semigroup in a group. The online version of the original article can be found under doi:.     

A note on Burgess’s estimate

By introducing the Rademacher-Menchov device, we prove “maximal” analogs of principal bounds of character sums. This allows us to present the Burgess method so as to separate the main idea of this method from the technical issues.     

A note on Barnette’s conjecture

A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture.     

A note on Gekeler’s h-function

For a subgroup L of the symmetric group \({S_{\ell}}\), we determine the minimal base size of \({GL_d(q) \wr L}\) acting on \({V_d(q)^{\ell}}\) as an imprimitive linear group. This is achieved by computing the number of orbits of GL_d(q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of V_m(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of \({GL_{d}(q) \wr L}\) satisfy a conjecture of Pyber concerning bases.     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

A note on Jeśmanowicz’ conjecture

Je?manowicz [9] conjectured that, for positive integers m and n with m > n, gcd(m,n) = 1 and \({m\not\equiv n\pmod{2}}\), the exponential Diophantine equation \({(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z}\) has only the positive integer solution (x, y, z) = (2, 2, 2). We prove the conjecture for \({2 \| mn}\) and m + n has a prime factor p with \({p\not\equiv1\pmod{16}}\).     

A note on Peano’s Theorem on time scales

In this paper we investigate the dynamic Cauchy problem in Banach spaces. We check how dense a time scale must be in such a way that Peano’s Theorem holds and we present a counterexample to Peano’s Theorem on a time scale with only one right dense point.     

A note on Linnik’s theorem on quadratic non-residues

Archiv der Mathematik - We present a short and purely combinatorial proof of Linnik’s theorem: for any $$varepsilon >0$$ there exists a constant $$C_varepsilon $$ such that for any...     

A note on Abel’s partial summation formula

Several applications of Abel’s partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated with a strong order unit, it is shown that the convergence of the series \(\sum x_{n}\) implies the convergence in density of the sequence \((nx_{n})_{n}\) to 0. This is done by extending the Koopman–von Neumann characterization of convergence in density. Also included is a new proof of the Jensen–Steffensen inequality based on Abel’s partial summation formula and a trace analogue of the Tomi?–Weyl inequality of submajorization.     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

A note on the selection of Benders’ cuts

A new cut selection criterion for Benders’ cuts is proposed and computationally analyzed. The results show that the new criterion is more robust—and often considerably faster—than the standard ones.     

A note on zeta functions of Ihara type for normal Kähler graphs

A regular Kähler graph is a compound of two regular graphs. When adjacency operators of component graphs are commutative, we introduce equivalence relations on sets of primitive bicolored paths, which are considered as sets of trajectory-segments of magnetic fields on this Kähler graph, we study their zeta functions of Ihara type, and show a correspondence to those for ordinary regular graphs.     

A note on Hardy’s theorem and rotations

Semigroup Forum - We give a very short proof, using the Hermite semigroup, to a generalized version of Hardy’s theorem due to Hogan and Lakey. We characterize $$fin L^2({mathbb {R}}^n)$$...     

A note on θ-tightness

In questo lavoro chiariamo il ruolo che la θ-tightness (introdotta dagli autori in [2]) ha nel determinare alcune estensioni di sottoinsiemi di uno spazio topologico.