Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

The Presentation Problem of the Conjugate Cone of the Hardy Space Hp (0 < p ≤ 1)

The Hardy space Hpis not locally convex if 0 < p < 1, even though its conjugate space(Hp) separates the points of Hp. But then it is locally p-convex, and its conjugate cone(Hp) p is large enough to separate the points of Hp. In this case, the conjugate cone can be used to replace its conjugate space to set up the duality theory in the p-convex analysis. This paper deals with the representation problem of the conjugate cone(Hp) p of Hpfor 0 < p ≤ 1, and obtains the subrepresentation theorem(Hp) p L∞(T, C p).     

The Domination Polynomial of a Graph at −1

Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial ${D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}$ , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at ?1. We give a construction showing that for each odd number n there is a connected graph G with D(G, ?1) = n.     

Rank p − 1 mod-p H-spaces

Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p ? 1. The first construction goes through when l = p ? 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give a practical test for when the criterion holds, and use this to give many new examples of finite H-spaces.     

The polynomial representation of the type An−1 rational Cherednik algebra in characteristic p | n

We study the polynomial representation of the rational Cherednik algebra of type A_n?1 with generic parameter in characteristic p for p | n. We give explicit formulas for generators for the maximal proper graded submodule, show that they cut out a complete intersection, and thus compute the Hilbert series of the irreducible quotient. Our methods are motivated by taking characteristic p analogues of existing characteristic 0 results.     

The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2

LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations $$\left\{ {(i_1 ,i_2 ,...,i_{n - 1} ,i_n ,),(i_n ,i_{n - 1} ,...,i_2 ,i_1 ,)} \right\}$$ is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n?2 geometric permutations. This upper bound is tight.     

A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

The region of attraction of the zero solution of y′(t)=−(y(t−1))2m+1

Estimates for the size of the region of attraction of the zero solution of y(t)=–(y(t–1))^2m+1 are given.     

Liouville-type results for solutions of −Δu=|u|p−1u on unbounded domains of RN

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $?\mathrm{\Delta }u={|u|}^{p?1}u$ on unbounded domains of ${\mathbb{R}}^N$ with $N?2$ and $p>1$. We prove some Liouville-type results and a classification theorem for $C^2$ solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation. To cite this article: A. Farina, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

From the Icosahedron to Natural Triangulations of??P 2 and S 2��S 2

We present two constructions in this paper: (a) a 10-vertex triangulation $\mathbb{C}P^{2}_{10}$ of the complex projective plane ?P ² as a subcomplex of the join of the standard sphere ( $S^{2}_{4}$ ) and the standard real projective plane ( $\mathbb{R}P^{2}_{6}$ , the decahedron), its automorphism group is A ₄; (b) a 12-vertex triangulation (S ²×S ²)₁₂ of S ²×S ² with automorphism group 2S ₅, the Schur double cover of the symmetric group S ₅. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S ²×S ². Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ?P ² has S ²×S ² as a two-fold branched cover; we construct the triangulation $\mathbb{C}P^{2}_{10}$ of ?P ² by presenting a simplicial realization of this covering map S ²×S ²???P ². The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S ²×S ², different from the triangulation alluded to in (b). This gives a new proof that Kühnel??s $\mathbb{C}P^{2}_{9}$ triangulates ?P ². It is also shown that $\mathbb{C}P^{2}_{10}$ and (S ²×S ²)₁₂ induce the standard piecewise linear structure on ?P ² and S ²×S ² respectively.     

Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.