The Cameron-Erd?s conjecture

A subset A of integers is said to be sum-free if a+b∉A for any a,b∈A. Let s(n) be the number of sum-free sets in interval [1,n] of integers. P. Cameron and P. Erd?s conjectured that s(n)=O(2^n/2). We show that for even n and for odd n, where are absolute constants, thereby proving the conjecture.     

Moments of Ramanujan?s generalized elliptic integrals and extensions of Catalan?s constant

We undertake a thorough investigation of the moments of Ramanujan?s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities.     

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.     

Combinatorial variations on Cantor?s diagonal

We discuss counting problems linked to finite versions of Cantor?s diagonal of infinite tableaux. We extend previous results of Brlek et al. (2004) [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative results follow and allow to look at the sub-class of the so-called bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type tableaux and coloring of hypergraphs having a square number of vertices.     

The Philamath’s Alphabet?WXYZ

Stamp CornerRobin Wilson

The Philamath’s Alphabet−WXYZ     

Thurston?s relative inequalities and Reeb components

We construct a family of codimension 1 foliations in a 3-manifold for which Thurston?s relative inequality holds, but for which the absolute one is violated. For this, we introduce a variant of these inequalities, which we call the relative^(±) inequality. Also we determine the class of foliations for which the relative^(±) inequality holds.     

On Quillen?s Theorem A for posets

A theorem of McCord of 1966 and Quillen?s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen?s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.     

An Integral Version of ?iri?’s Fixed Point Theorem

We establish a new fixed point theorem for mappings satisfying a general contractive condition of integral type. The presented theorem generalizes the well known Ćirić’s fixed point theorem [Lj. B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. 12 (26) (1971) 19-26]. Some examples and applications are given.     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

Lelek?s problem is not a metric problem

We show that Lelek?s problem on the chainability of continua with span zero is not a metric problem: from a non-metric counterexample one can construct a metric one.     

Revisiting the foundations of Barbilian?s metrization procedure

In the present work we prove that one of Barbilian?s theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be described by Barbilian?s metrization principle. Secondly, that all the Riemannian metric corresponding to these distances can be obtained with the same unique procedure derived from the main theorem in the text (Theorem 2.5). We show how the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in the same way using Theorem 2.5, which appears here for the first time and is an extension of a Barbilian classical result (Barbilian, 1960 [7]). Furthermore, we obtain metrics corresponding to quadratic forms with signature that includes minus. By considering the norms provided by either Lorentz or Minkowski (pseudo-)inner product as influence functions, two oscillant distances can be generated in some subsets of Lorentz or Minkowski plane. The extension of 1960 Barbilian?s theorem mentioned above allow us to obtain the metrics attached to these two Barbilian distances on corresponding subsets of Lorentz and Minkowski 2-dimensional spaces. The geometric study concludes that these metrics are generalized Lagrange metrics. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance is also proved.     

Regge?s Einstein-Hilbert functional on the double tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge?s Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.     

Computing Perelman?s ν-functional

This is a short note in which we show how to calculate the value of Perelman?s ν-functional for a variety of metrics. In particular we complete the calculation of values for the known 4-dimensional Einstein and shrinking Ricci soliton metrics.     

An approximate version of Sumner?s universal tournament conjecture

Sumner?s universal tournament conjecture states that any tournament on 2n−2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on (2+o(1))n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed Δ, any tournament on (1+o(1))n vertices contains a copy of any directed tree on n vertices with maximum degree at most Δ.     

A simple generalization of Ger?gorin��s theorem

It is well known that the spectrum of a given matrix A belongs to the Ger?gorin set ??(A), as well as to the Ger?gorin set applied to the transpose of A, ??(A ^T ). So, the spectrum belongs to their intersection. But, if we first intersect i-th Ger?gorin disk ?? _i (A) with the corresponding disk $\Gamma_i(A^T)$ , and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set ??(A)??????(A ^T ). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.     

Hensley?s problem for complex and non-Archimedean meromorphic functions

Büchi?s problem asks if there exists a positive integer M such that all x₁,…,xM∈Z satisfying the equations for all 3?r?M must also satisfy for some integer x. Hensley?s problem asks if there exists a positive integer M such that, for any integers ν and a, if ²(ν+r)−a is a square for 1?r?M, then a=0. It is not difficult to see that a positive answer to Hensley?s problem implies a positive answer to Büchi?s problem. One can ask a more general version of the Hensley?s problem by replacing the square by n-th power for any integer n?2 which is called the Hensley?s n-th power problem. In this paper we will solve Hensley?s n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.     

A conjecture of Erd?s on graph Ramsey numbers

The Ramsey number r(G) of a graph G is the minimum N such that every red–blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics.One of the oldest results in Ramsey Theory, proved by Erd?s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most . Motivated by this estimate Erd?s conjectured, more than a quarter century ago, that there is an absolute constant c such that for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.