Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the n-sphere, with n 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existence of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.Mathematics Subject Classification (2000): 35J60, 53C21, 58J05Send offprint requests to: Khalil El Mehdi     

Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space

Let G_m,n be the Green function of with Dirichlet boundary conditions We establish some estimates on G_m,n, including a 3G-Theorem. Next, we introduce a Kato class of functions and we exploit properties of these functions to study the existence of positive solutions of some m-polyharmonic nonlinear elliptic problems.Mathematics Subject Classification (2000): 34B27, 35J40     

On the integrability of the <Emphasis Type="Italic">n</Emphasis>-centre problem

It is known that for n3 centres and positive energies the n-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary g>1 independent integrals of Gevrey class g exist, no real-analytic (that is, Gevrey class 1) independent integral exists.Mathematics Subject Classification (2000): 70F10, 37J30, 37J35, 37N05, 70F15, 70H06, 81U10     

On the sectional geometric genus of quasi-polarized varieties,II

Let (X,L) be a quasi-polarized variety of dim X=n. In the previous paper we gave a new invariant (the i-th sectional geometric genus) of (X,L), which is a generalization of the degree and the sectional genus of (X,L). In this paper we study some properties of the sectional geometric genus, and we consider the i-th sectional geometric genus of some special varieties. This research was partially supported by Grant-in-Aid for Scientific Research (No.14740018) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject Classification (2000):Primary 14C20; Secondary 14J30, 14J32, 14J35, 14J40, 14J45, 14N99     

The Brunn-Minkowski inequality for <Emphasis Type="Italic">p</Emphasis>-capacity of convex bodies

In this paper we prove the Brunn-Minkowski inequality for the p-capacity of convex bodies (i.e convex compact sets with non-empty interior) in R ⁿ , for every p(1,n). Moreover we prove that the equality holds in such inequality if and only if the involved bodies coincide up to a translation and a dilatation. Mathematics Subject Classification (2000):35J60, 31B15, 39B62, 52A40     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

On the structure of graphs with a unique k‐factor

In this paper we study the structure of graphs with a unique k‐factor. Our results imply a conjecture of Hendry on the maximal number m (n,k) of edges in a graph G of order n with a unique k‐factor: For we prove and construct all corresponding extremal graphs. For we prove . For n = 2kl, l ∈ ℕ, this bound is sharp, and we prove that the corresponding extremal graph is unique up to isomorphism. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 227–243, 2000     

Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arborescence of order n > 1. This is a particular case of Sumner's conjecture: every tournament of order 2n − 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n − 5)/2 contains every oriented tree of order n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244–256, 2000     

On the second sectional H-arithmetic genus of polarized manifolds

Let (X,L) be a polarized variety defined over the complex number field with dim X=n. In this paper we introduce the notion of the i-th sectional H-arithmetic genus ^H_i(X,L) for every integer i with 0in. We expect that this invariant has a property similar to the Euler-Poincaré characteristic of the structure sheaf of i-dimensional varieties. In this paper, we consider the case where X is smooth and i=2, and we study a polarized version of some results in the theory of surfaces.Mathematics Subject Classification (2000): 14C20, 14C17, 14C40, 14J29, 14J30, 14J35, 14J40This research was partially supported by the Grant-in-Aid for Young Scientists (B) (No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications

We study some generalized small Lebesgue spaces and their associated Sobolev spaces. In particular, we prove that small Lebesgue-Sobolev spaces W^1,(p(Ω) are compactly embedded in , p < n. As an application, we study variational problems involving critical exponents under multiple constraints. Mathematics Subject Classification (2000) 46E30, 46E35, 46B70, 26D07, 35J60.     

A lower bound for the crossing number of Cm × Cn

We prove that the crossing number of C_m × C_n is at least (m − 2)n/3, for all m, n such that n ≥ m. This is the best general lower bound known for the crossing number of C_m × C_n, whose exact value has been long conjectured to be (m − 2)n, for 3 ≤ m ≤ n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 222–226, 2000     

Some existence results for the Webster scalar curvature problem in presence of symmetry

We prove some existence results for the Webster scalar curvature problem on the Heisenberg group and on the unit sphere of ⁿ⁺¹, under the assumption of some natural symmetries of the prescribed curvatures. We use variational and perturbation techniques. Mathematics Subject Classification (2000) 35J20, 35H20, 35J60, 43A80     

The Ramsey number for a cycle of length five vs. a complete graph of order six

It has been conjectured that r(C_m, K_n) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 4. This has been proved recently for n = 4 and n = 5. In this paper, we prove that r(C₅, K₆) = 21. This raises the possibility that r(C_m, K₆) = 5m − 4 for all m ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 99–108, 2000     

More on the existence of small quasimultiples of affine and projective planes of arbitrary order

The functions a(n) and p(n) are defined to be the smallest integer λ for which λ‐fold quasimultiples affine and projective planes of order n exist. It was shown by Jungnickel [J. Combin. Designs 3 ( 6 ), 427–432] that a(n),p(n) < n¹⁰ for sufficiently large n. In the present paper, we prove that a(n),p(n) < n³. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 182–186, 2001     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let and suppose that f : K ⁿ →K ⁿ is nonexpansive with respect to the l ₁-norm, , and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.     

On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ(n^3.5logn); but the best combinatorial algorithms have approximation ratio 4/5 only, achieved in O(n²) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O(n²) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.     

The crossing number of K1,3,n and K2,3,n

In this article, we will determine the crossing number of the complete tripartite graphs K_1,3,n and K_2,3,n. Our proof depends on Kleitman's results for the complete bipartite graphs [D. J. Kleitman, The crossing number of K_5,n. J. Combinatorial Theory 9 (1970) 315-323].     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009