On the best constant for the Friedrichs-Knapp-Stein inequality in free nilpotent Lie groups of step two and applications to subelliptic PDE

In this article we propose to find the best constant for the Friedrichs-Knapp-Stein inequality in F_2n,2, that is the free nilpotent Lie group of step two on 2n generators, and to prove the second-order differentiability of subelliptic p-harmonic functions in an interval of p.     

Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent

In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear part, we can obtain the existence of multiple solutions for the equations. The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015).     

The minimum degree threshold for perfect graph packings

Let H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H,n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G)≥k contains a perfect H-packing. We show that

Quasi-randomness and the distribution of copies of a fixed graph

We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles. A similar improvement of [10] is also obtained for any fixed graph other than the triangle, and for any edge density other than 1/2. The proof relies on a theorem of Gottlieb [7] in algebraic combinatorics, concerning the rank of set inclusion matrices.     

Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions. This paper was completed while the first author was visiting the University of Aveiro as an Invited Scientist. The hospitality and financial support of the host institution are gratefully acknowledged. The second and third authors acknowledge the partial financial support of the Portuguese Foundation for Science and Technology (FCT) under the research project POCI/MAT/55524/2004.     

Perturbation from symmetry for indefinite semilinear elliptic equations

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) =  ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).     

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H ₂-convex function in vanishing on ∂Ω one has

Special matchings and Coxeter groups

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.] All authors are partially supported by EU grant HPRN-CT-2001-00272. Received: 30 October 2006     

Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ  × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ  ≥ H ≥ a ₂ H _δ for some a ₁, a ₂ > 0 where

A measurable-group-theoretic solution to von Neumann’s problem

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors. D. Gaboriau’s research was supported by CNRS. R. Lyons’ research was supported partially by NSF grant DMS-0705518 and Microsoft Research.     

Birth control for giants

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v ₁, v ₂}, {v ₃, v ₄} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t _c so that S(t) → ∞ as t approaches t _c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t _c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t _c we show the existence of a giant component, so that t = t _c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component. Research supported by the Australian Research Council, the Canada Research Chairs Program and NSERC. Research partly carried out while the author was at the Department of Mathematics and Statistics, University of Melbourne.     

Unbounded functional calculus for bounded groups with applications

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines. M. Kovács is partially supported by postdoctoral grant No. 623-2005-5078 of the Swedish Research Council (VR).     

<Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free subgraphs of random graphs revisited

In Combinatorica 17(2), 1997, Kohayakawa, ?uczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G _{n, p} . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K ₄, thereby providing a conceptually simple proof for the main result in the paper cited above.     

Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form

The mixed problem in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> for some two-dimensional Lipschitz domains

We consider the mixed problem,

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

Pro-p groups of rank 3 and the question of Iwasawa

For a prime p > 3 we determine all pro-p groups that satisfy d(G) = d(H) = 3 for all open subgroups H of G. Received: 29 July 2008     

Product set estimates for non-commutative groups

We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre. T. Tao is supported by a grant from the Packard Foundation.     

A Martinelli-Bochner formula on fractal domains

A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in [5]. On the other hand, the Cauchy transform associated to Clifford analysis has been involved recently with fractional metric dimensions and fractals, see [1, 2, 3]. In this paper we consider the question of possible generalizations of the Cauchy integral formula to domains with fractal boundary. As an application, we prove a Martinelli-Bochner type formula for several complex variables on such pathological domains. The proof makes heavy use of the isotonic approach of the monogenic functions theory. Received: 8 October 2008