On the Cohomology Sequence in a Semiabelian Category

In a semiabelian category, a strictly exact sequence 0→A→B→C→0 of cochain complexes gives rise to the cohomology sequence ...→H ⁿ(A) →H ⁿ(B)→ H ⁿ(C)→ H ⁿ⁺¹ (A) →.... We study conditions for exactness of the homology sequence at a given term.     

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

On Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G ₁ → G ₀ → G ⁰ → G ¹ → … such that the complex Hom _R (G, H) is exact for each projective left R-module H, the module Im(G ₀ → G ⁰) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.     

The limit behavior of the second‐grade fluid equations

In the paper, the limit behavior of solutions to the second‐grade fluid system with no‐slip boundary conditions is studied as both ν→0 and α→0. More precisely, it is verified that the convergence from second‐grade fluid system to Euler system holds as ν→0 and α→0 independently under the radial symmetry case.     

Approximation of Solutions of Nonlinear Equations of Monotone and Hammerstein Type

Let X be a real uniformly smooth and uniformly convex Banach space with dual X ^*. Let A: X → X ^* be a bounded uniformly submonotone map. It is proved that a Mann-type approximation sequence converges strongly to Jx ^* where x ^* ∈ N(A). Furthermore, as an application of this result an iterative sequence which converges strongly to a solution of the Hammerstein equation u+KFu = 0 is constructed where, F:X→X^* and K:X^*→X are monotone-type mappings. No invertibility assumption is imposed on K. Moreover, neither K nor F need be compact. Finally, our method is of independent interest.     

Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two

We consider the blowup solution ( u,n,v )( t ) of the Zakharov equations where u : R ² → C, n : R ² → R, v: R² → R² in the energy space H₁ = {(u,n,v) η H¹ × L² × L²}. We show that there is a constant c depending on the L²-norm of u₀ such that where T is the blowup time. We check that this estimate is optimal and give further applications. © 1996 John Wiley & Sons, Inc.     

Acyclic and oriented chromatic numbers of graphs

The oriented chromatic number χ_o(G ^→) of an oriented graph G ^→ = (V, A) is the minimum number of vertices in an oriented graph H ^→ for which there exists a homomorphism of G ^→ to H ^→. The oriented chromatic number χ_o(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for χ_o(G) in terms of χ_a(G). An upper bound for χ_o(G) in terms of χ_a(G) was given by Raspaud and Sopena. We also give an upper bound for χ_o(G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. © 1997 John Wiley & Sons, Inc.     

On slightly P-coherent rings

A ring R is called left slightly P-coherent if C is P-injective, for every left R-module exact sequence 0→A→B→C→0 with A and B P-injective. The properties of slightly P-coherent rings and several examples are studied to show that left slightly P-coherent rings fall in between left P-coherent rings and left strongly P-coherent rings. In terms of some derived functors, some homological dimensions over these rings are investigated. As applications, some new characterizations of p.p.rings are given.     

Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well-posed problem (in the Hadamard classical sense) in spaces H^k(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (a_n, ?_n) → (a, ?) in H^k × H^k and if f_n → f in ??²(0, T;H^k), then (v_n, g_n) → (v, g) in ?? (0, T; H^k × H^k). The method followed here (see also [8]) also applies to the non-barotropic case p = p(ρ, s) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural-stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

Anti‐periodic solutions for evolution equations with mappings in the class (S+)

In this paper, we study the existence of anti‐periodic solutions for the first order evolution equation in a Hilbert space H, where G : H → ? is an even function such that ?G is a mapping of class (S₊) and f : ? → ? satisfies f(t + T) = –f(t) for any t ∈ ? with f(·) ∈ L²(0, T; H). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coherent Rings and Gorenstein FP-Injective Modules

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E ₁ → E ₀ → E ⁰ → E ¹ → … of FP-injective left R-modules with M = ker(E ⁰ → E ¹) such that Hom _R (P, ?) leaves the sequence exact whenever P is a finitely presented left R-module with pd _R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules.     

Cleavages of graphs: the spectral radius

A cleavage of a finite graph G is a morphism f : H → G of graphs such that if P is the m × n characteristic matrix defined as P _ik = 1 if i ∈ f ^?1(k), otherwise = 0, then A(H)P ≤ PA(G), where A(G) and A(H) are the adjacency matrices of G and H, respectively. This concept generalizes induced subgraphs, quotients of graphs, Galois covers, path-tree graphs and others. We show that for spectral radii we have the inequality ρ(H) ≤ ρ(G). Equality holds only in case f : H → G is an equivariant quotient and H has isoperimetric constant i(H) = 0.     

On the Degeneracy Locus of a Map of Vector Bundles on Grassamannian Varieties

Let ℒ︁ be a line bundle on a smooth curve C of genus g ≥ 2 and let W ⊂ H⁰ (ℒ︁) be a subspace of dimension r +1, in this paper we study the natural map μ_W : W ⊗ H⁰ (ω_C) → H⁰ (ℒ︁ ⊗ ω_C). Let D ⊂ G(r + 1, H⁰(ℒ︁)) be the locus where μ_W fails to be surjective: we prove that, if C is not hyperelliptic of genus g ≥ 3, D is an irreducible and reduced divisor on G(r + 1, H⁰(ℒ︁)) for any r ≥ 3, and for r = 2 if the curve C is not trigonal.     

Asymptotic behaviour of curved rods by the unfolding method

We consider in this work general curved rods with a circular cross‐section of radius δ. Our aim is to study the asymptotic behaviour of such rods as δ→0, in the framework of the linear elasticity according to the unfolding method. It consists in giving some decompositions of the displacements of such rods, and then in passing to the limit in a fixed domain. A first decomposition concerns the elementary displacements of a curved rod which characterize its translations and rotations, and the residual displacements related to the deformation of the cross‐section. The second decomposition concerns the displacements of the middle‐line of the rod. We prove that such a displacement can be written as the sum of an inextensional displacement and of an extensional one. An extensional displacement will modify the length of the middle‐line, while an inextensional displacement will not change this length in a first approximation. We show that the H¹‐norm of an inextensional displacement is of order 1, while that of an extensional displacement is in general, of order δ. A priori estimates are established and convergence results as δ→0, are given for the displacements. We give their unfolded limits, as well as the unfolded limits of the strain and stress tensors. To prove the convergence of the strain tensor, the introduction of elementary and residual displacements appears as essential. By passing to the limit as δ→0 in the linearized system of the elasticity, we obtain on the one hand, a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacements and the limit of the angle of torsion. Copyright © 2004 John Wiley & Sons, Ltd.     

Volumes Spanned by Random Points in the Hypercube

Consider the hypercube [0, 1]ⁿ in Rⁿ. This has 2ⁿ vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let V_n be its expected volume. How large should N(n) be to pick up significant volume? Let k=2/√≈1.213, and let ? > 0. We shall show that, as n→∞, V_n→0 if N(n)?(k??)ⁿ →1 if N(n) ? (k + ?)ⁿ. A similar result holds for sampling uniformly from within the hypercube, with constant .     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

BLOWING UP PROJECTIVELY NORMAL SPACE CURVES AND VERY AMPLE DIVISORS

Let C ? P ³ be a projectively normal curve and π : X (C) → P ³ the blowing-up of C,E the exceptional divisor of π and H ∈(X (C)) the total transform of the hyperplane line bundle. Let H (C,t) be the Hilbert function of C and Δ² H its second difference function. Let σ be the minimal integer such that Δ² H (C σ) = 0. Here we prove that is C has no σ-secant line, then |σ H – E| is very ample.     

The Structure Theorem of Weak Comodule Algebras

This article mainly gives the structure theorem of weak comodule algebras, that is, assume that H is a weak Hopf algebra, and B a weak right H-comodule algebra, if there exists a morphism φ: H → B of a weak right H-comodule algebras, then there exists an algebra isomorphism: B ? B ^coH #H, where B ^coH denotes the coinvariant subalgebra of B, and B ^coH #H denotes the weak smash product.