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 Chowʼs entropy functional for the Gauss curvature flow

Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional that is monotone along the unnormalized flow and whose critical point is a shrinking self-similar solution.     

A note on two Erdősʼs proofs of the infinitude of primes

P. Erdős found numerous theorems, problems, results and conjectures in elementary number theory. Some of them are two Erdősʼs proofs of of the famous Euclidʼs theorem on the infinitude of primes. As noticed below, one of these proofs immediately implies the fact that the number of primes smaller than x is $⩾\log x/(2\log 2)$.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

Irreducibility and extensions of Ostrowski’s Theorem

In this paper an extension of Ostrowski’s Theorem for complex square irreducible matrices is presented. Also extensions of similar statements for square complex matrices are analyzed and completed. Most of the statements in this work cover also the case of reducible matrices.     

Nakajimaʼs remark on Hennʼs proof

We fill up a gap in Hennʼs proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic.     

A remark on Gagola’s Theorem

In this paper, we generalize Gagola’s Theorem [1]. Firstly we obtain several new identities. With the help of these identities, we prove a conclusion similar with Gagola’s under some more general conditions. Finally, we get a result regarding the control of p-transfer by Tate’s Theorem.     

A Note on Schnyder’s Theorem

We give an alternate proof of Schnyder’s Theorem, that the incidence poset of a graph G has dimension at most three if and only if G is planar.     

A note on Rotfel’d Theorem in a sector

The main purpose of this work is to present a better bound than the recent result given by Zhang [Linear Multilinear Algebra. 2015;63:2511–2517].     

A note on Thomassen?s conjecture

In 1983 C. Thomassen [8] conjectured that for every k,g∈N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. A result of Pyber, Szemerédi, and the second author implies that the conjecture is true for every graph G with average .We strengthen this and show that the conjecture holds for every graph G with average for some constants α, β depending on k and g.     

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.     

Ramanujanʼs Master Theorem for Riemannian symmetric spaces

Ramanujan?s Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujan?s Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one.