Conditions for dominated splittings

In this paper, we solve the problem proposed by Lan Wen for the case of dimM = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P＊i（f）.     

Codimension one generic homoclinic classes with interior

We study C ¹-generic diffeomorphisms with a homoclinic class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

The Lyapunov exponents of C 1 hyperbolic systems

Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure μ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of μ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

On the existence of convex decompositions of partially separable functions

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsf _i whose Hessians have nontrivial nullspacesN _i , Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionf _i is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf _i such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureN _i ¹ have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachN _i is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDL^T factorization without fill-in.     

Korovkin-type theorems and applications

Let {T _n } be a sequence of linear operators on C[0,1], satisfying that {T _n (e _i )} converge in C[0,1] (not necessarily to e _i ) for i = 0,1,2, where e _i = t ⁱ . We prove Korovkin-type theorem and give quantitative results on C ²[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.      

On Strongly Copure Flat Modules and Copure Flat Dimensions

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext ⁱ (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Korovkin-type theorem and application

Let (L_n) be a sequence of positive linear operators on C[0,1], satisfying that (L_n(e_i)) converge in C[0,1] (not necessarily to e_i) for i=0,1,2, where e_i(x)=xⁱ. We prove that the conditions that (L_n) is monotonicity-preserving, convexity-preserving and variation diminishing do not suffice to insure the convergence of (L_n(f)) for all fC[0,1]. We obtain the Korovkin-type theorem and give quantitative results for the approximation properties of the q-Bernstein operators B_n,q as an application.     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

Paths,cycles, and arc-connectivity in digraphs

In this paper we prove the following theorem: Let D be a k-arcconnected digraph (multiple arcs allowed). If x is a vertex of D and / is an integer with / ≤ k, then for any / disjoint arc pairs {f₁, g₁}, ?, {f₁, g₁}, where f₁, ?, f₁ are arcs with head at x and g₁, ?, g₁ are arcs with tail at x, there exist in D / arc-disjoint cycles C₁, ?, C₁ such that {f_i, g_i} ? E(C_i) for each i (E(C_i) denotes the arc set of C_i) and such that D - ∪ E(C_i) is (k - 1)-arc-connected. Several interesting results are deduced from this theorem. Our results generalize the early works of Mader (“On a Property of n-Edge-Connected Digraphs,” Combinatorica, vol. 1 [1981], pp. 385-386) and Shiloach (“Edge-Disjoint Branching in Directed Multigraphs,” Information Processing Letters, vol. 8 [1979], pp. 24-27). An extension of Mader's theorem about admissible liftings of digraphs is also obtained in this paper. © 1995 John Wiley & Sons, Inc.     

Extending partial isomorphisms on finite structures

We prove the following theorem:Let A be a finite structure in a fixed finite relational language,p ₁,...,p _m partial isomorphisms of A. Then there exists a finite structure B, and automorphismsf _i of B extending thep _i 's. This theorem can be used to prove the small index property for the random structure in this language. A special case of this theorem is, if A and B are hypergraphs. In addition we prove the theorem for the case of triangle free graphs.     

A criterion for the primeness of ideals generated by polynomials with separated variables

Proving primeness of an idealI=〈f ₁, …,f _m〉 in a polynomial ringR=K[X ₁, …,X _n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef _iare large. In this paper we will give an easy criterion for the primeness ofI if thef _iare polynomials with separated variables, i.e. no mixed monomials occur in thef _i. The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University.     

Cohen-Macaulay, shellable and unmixed clutters with a perfect matching of König type

Let C be a clutter with a perfect matching e₁,…,e_g of König type and let Δ_C be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show that Δ_C is pure shellable. We are able to describe, in combinatorial and algebraic terms, when Δ_C is pure. If C has no cycles of length 3 or 4, then it is shown that Δ_C is pure if and only if Δ_C is pure shellable (in this case e_i has a free vertex for all i), and that Δ_C is pure if and only if for any two edges f₁,f₂ of C and for any e_i, one has that f₁∩e_i⊂f₂∩e_i or f₂∩e_i⊂f₁∩e_i. It is also shown that this ordering condition implies that Δ_C is pure shellable, without any assumption on the cycles of C. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if C is admissible and complete, then C is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi-on the structure of unmixed simplicial trees-to clutters with the König property without 3-cycles or 4-cycles.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

C1, 1 Regularity in semilinear elliptic problems

In this paper we give an astonishingly simple proof of C^{1, 1} regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optical results. The setting we'll consider is the following. Let u be a solution to where B, is the unit ball in ?ⁿ, f(x, t) is a bounded Lipschitz function in x, and f_t′ is bounded from below. Then we prove that u ? C^{1, 1} (B_1/2). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig. © 2002 Wiley Periodicals, Inc.     

Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces

Let M be a two-dimensional complex manifold and let be a holomorphic map that fixes pointwise a (possibly) singular, compact, reduced and globally irreducible curve . We give a notion of degeneracy of f at a point of C. It turns out that f is non-degenerate at one point if and only if it is non-degenerate at every point of C. When f is non-degenerate on C, we define a residual index for f at each point of C. Then we prove that the sum of the indices is equal to the self-intersection number of C. Received: 15 May 2000; in final form: 10 July 2001 / Published online: 1 February 2002     

ON ADJUNCTIONS INDUCING THE SAME MONAD

ABSTRACT

Consider an adjunction <F,U;n,c>: K. → A, T = <T,n,u> the monad it induces in K and ø: A → K^T the comparison functor, K^T being the category of T-algebras. By ø*: Proj U → Proj U^T we denote the restriction and co-restriction of ø to the subcategories of U-projective and U^T -projective objects, respectively. In this paper we deal with the following problem, raised by R.-E. Hoffmann in [5] 1.16 (b):

Assuming that ø* is an equivalence of categories when is it possible to find a category C and a right adjoint functor V: C → K inducing the same monad T in K, and a full reflective embedding E: A → K, such that:

(1) V.E = U.

(2) ø = ø'. E for the comparison functor ø': C → K^T .

(3) F'X is contained (via E) in A, for each K-object X, F' being the left adjoint of V.

(4) ø': C → K^T has a full and faithful left adjoint L'.

We prove that there exists a pair (C,V) satisfying the conditions of the problem, with A an isomorphism-closed subcategory of C, such that:

(5) For all C ? Obj C the reflection map r_C: C → A is ø'-initial.

We also prove that this pair (C,V) is the universal solution satisfying condition (5), i.e. if (C_i,V_i) is a pair satisfying conditions (1)-(5) with E_i: A → C₂ the embedding and L_i left adjoint to the comparison functor ø_i: C_i K^T then there exists a unique full and faithful functor H_i: C → C_i such that H. E = E_i and H_i. L'—L_i. Moreover the universal solution is uniquely determined up to isomorphisms of categories and natural isomorphisms of functors. Finally, we study a particular situation and find, within the solutions of the problem satisfying two further conditions, the lease and the largest element. We conclude the paper with an example of this situation.     

A simple proof for persistence of snap-back repellers

In this article, we show that if f has a snap-back repeller then any small C¹ perturbation of f has a snap-back repeller, and hence has Li-Yorke chaos and positive topological entropy, by simply using the implicit function theorem. We also give some examples.