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 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 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 Bridgeland’s Hall algebra of two-periodic complexes

We show that the Hall algebra of two-periodic complexes, which is recently introduced by T. Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.     

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 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 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 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 the equilibria of the unbounded traveler’s dilemma

When there is no upward limit on admissible claims, the traveler’s dilemma admits a continuum of symmetric mixed strategy equilibria in addition to the pure strategy equilibrium in which both players ask and obtain the minimum. The payoff of any of these equilibria exceeds the payoff of the pure strategy one and any claim represents an attainable payoff. If the distinction between a large and an unbounded action set is fuzzy, this result can explain some puzzling stylized facts on the behavior of experimental subjects in the game.     

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 q-difference equations for Ramanujan’s integrals

The Ramanujan Journal - This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to...     

A note on Alberti’s Luzin-type theorem for gradients

Ricerche di Matematica - We give a “soft” proof of Alberti’s Luzin-type theorem in Alberti (J Funct Anal 100:110–118, 1991), using elementary geometric measure theory and...