Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the ε-entropy

We prove a conjecture of Zahariuta which itself solves a problem of Kolmogorov on the -entropy of some classes of analytic functions. For a given holomorphically convex compact subset K in a pseudoconvex domain D in Cⁿ, Zahariutas conjecture consists in approximating the relative extremal function u^*_K,D, uniformly on any compact subset of DK, by pluricomplex Green functions on D with logarithmic poles in the compact subset K.     

Proof of Varagnolo–Vasserot conjecture on cyclotomic categories $${\mathcal {}}$$

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category \({\mathcal {O}}\) for a cyclotomic rational Cherednik algebra and a suitable truncation of an affine parabolic category \({\mathcal {O}}\) that, in particular, implies Rouquier’s conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier’s deformation approach as well as categorical actions on highest weight categories and related combinatorics.     

Proof of Andrews' conjecture on a 4φ3 summation

We give a new proof of a ₄φ₃ summation due to G.E. Andrews and confirm another ₄φ₃ summation conjectured by him recently. Some variations of these two ₄φ₃ summations are also given.     

Decimation of the Dyson–Ising ferromagnet

We study the decimation to a sublattice of half the sites of the one-dimensional Dyson–Ising ferromagnet with slowly decaying long-range pair potentials of the form $\frac{1}{{|i?j|}^{\alpha }}$, deep in the phase transition region ($1<\alpha \le 2$ and low temperature). We prove non-Gibbsianness of the decimated measures at low enough temperatures by exhibiting a point of essential discontinuity for the (finite-volume) conditional probabilities of decimated Gibbs measures. This result complements previous work proving conservation of Gibbsianness for fastly decaying potentials ($\alpha >2$) and provides an example of a “standard” non-Gibbsian result in one dimension, in the vein of similar results in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.     

Around the Proof of the Legendre–Cauchy Lemma on Convex Polygons

We briefly describe the history of the proofs of the well-known Cauchy lemma on comparison of the distances between the endpoints of two convex open polygons on a plane or sphere, present a rather analytical proof, and explain why the traditional constructions lead in general to inevitable appearance of nonstrictly convex open polygons. We also consider bendings one to the other of two isometric open or closed convex isometric polygons.     

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.     

Solving the Ku–Wales conjecture on the eigenvalues of the derangement graph

We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue decreases whenever the corresponding partition with a fixed first part decreases in the dominance order. In particular, this settles affirmatively a conjecture of Ku and Wales [C.Y. Ku, D.B. Wales, Eigenvalues of the derangement graph, J. Combin. Theory Ser. A 117 (2010) 289–312] regarding the lower and upper bounds for the absolute values of these eigenvalues.     

Proof of the Erdős–Faudree Conjecture on Quadrilaterals

In this paper, we prove the Erdős–Faudree’s conjecture: If G is a graph of order 4k and the minimum degree of G is at least 2k then G contains k disjoint cycles of length 4.     

Proof of the bandwidth conjecture of Bollobás and Komlós

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r ? 1)/r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most β n and maximum degree at most Δ, then G contains a copy of H.     

Proof of the Erdős matching conjecture in a new range

Let s > k ≧ 2 be integers. It is shown that there is a positive real ε = ε(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + ε) every k-graph on n vertices with no more than s pairwise disjoint edges has at most \(\left( {\begin{array}{*{20}{c}} {\left( {s + 1} \right)k - 1} \\ k \end{array}} \right)\) edges in total. This proves part of an old conjecture of Erd?s.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

A conjecture on the Eichler cohomology of automorphic forms

We prove partially a conjecture of Knopp about the Eichler cohomology of automorphic forms on H-groups.     

Towards the Eaton-Moretó conjecture on the minimal height of characters

Abstract

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ? > 5 no ?-block of any quasi-simple group can be a minimal counterexample.     

Curves on threefolds and a conjecture of Griffiths–Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.     

Proof of a conjecture of José L. Rubio de Francia

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1<p<∞, the characteristic function χ_I of an interval I in Γ is a p—multiplier with a uniform bound (independent of I) on the corresponding operator S_I on L^p(G). In this note it is shown that, for 1<p,q<∞, there is a constant C_p,q, independent of G and the particular ordering on Γ, such that for all sequences {I_j} of intervals in Γ and all sequences {f_j} in L^p(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when The present proof uses a transference argument, an approach which shows that any constant C_p,q for which the inequality holds when G = will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space.The work of the first author was partially supported by a grant from the National Science Foundation (U.S.A.). The second and third authors were partially supported by the HARP network HPRN-CT-2001-00273 of the European Commission and by grant BFM2001-0188 of Ministerio de Ciencia y Tecnologia.     

Proof of Ding’s conjecture on maximal stable sets and maximal cliques in planar graphs

X. Deng et al. proved Chvātal＇s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvátal＇s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.