Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Finite domination, Novikov homology and nonsingular closed 1-forms

Let X be a finite connected CW-complex and ρ: a regular covering space with free abelian covering transformation group. For ξ ∈ H¹ (Xℝ) we define the notion of ξ-contractibility of X. This notion is closely related to the vanishing of the Novikov homology of the pair (X,ξ). We show that finite domination of is equivalent to X being ξ-contractible for every nonzero ξ with ρ*ξ =0  ∈ H¹(; ℝ). If M is a closed connected smooth manifold the condition that M is ξ-contractible is necessary for the existence of a nonsingular closed 1-form representing ξ. Also ξ-contractibility guarantees the definition of the Latour obstruction τ_L(M,ξ) whose vanishing is then sufficient for the existence of a nonsingular closed 1-form provided  dim M≥6. Now if ρ: is a finitely dominated regular ℤ^k-covering space we get that τ_L(M,ξ) is defined for every nonzero ξ with ρ*ξ=0 and the vanishing of one such obstruction implies the vanishing of all such τ_L(M,ξ).     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Some q-convexity properties of coverings of complex manifolds

Let C be a nowhere dense compact analytic subset of a 2-dimensional complex manifold X. A result of Napier is the construction of a neighborhood V of C such that for any covering space in which is holomorphically convex, there is a C ^∞ exhaustion function φ on which is strictly plurisubharmonic on π^-1(V) away from the compact irreducible components of . Colţoiu and Vajaitu obtained a q-convex version of this result for C a q-dimensional fiber of a proper surjective submersion X→Y. The goal of this paper is a similar version for Cprojective.     

Non-isogenous superelliptic Jacobians

Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set of roots of f and that Gal(h) acts doubly transitively on as well. Let J(C_f,ℓ) and J(C_h,ℓ) be the Jacobians of the superelliptic curves C_f,ℓ:y^ℓ=f(x) and C_h,ℓ:y^ℓ=h(x) respectively. We prove that J(C_f,ℓ) and J(C_h,ℓ) are not isogenous over K_a if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity.     

On the Rellich inequality with magnetic potentials

In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C₀^∞(Rⁿ \{0}) and n≠2where the constant C(n):=n²(n−4)²/16 is sharp. For n=2 extra conditions were required for f, and for n=4, C(4)=0, producing a trivial inequality. Influenced by recent work of Laptev-Weidl on Hardy-type inequalities in R², the authors show that for n≥2, the inclusion of a magnetic field B=curl(A) of Aharonov-Bohm type yields non-trivial Rellich-type inequalities of the formwhere Δ_A=(∇−iA)² is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant C(n,α) depends upon the distance of the magnetic flux to the integers Z. When the flux is an integer and α=0, the inequalities reduce to Rellich’s inequality.The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.     

Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

Existence of singular solutions of a degenerate equation in R 2

Let a₁,a₂, . . . ,a_m ∈ ℝ², 2≤f ∈ C([0,∞)), g_i ∈ C([0,∞)) be such that 0≤g_i(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation u_t=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−a_i|→−g_i(t) as |x−a_i|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u₀ where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ² with prescribed singularities at a finite number of points in the domain.     

Versal embeddings of compact 3-pseudoconcave CR submanifolds

We prove that for an induced CR structure on a compact, generic, regular 3-pseudoconcave CR submanifold of a complex manifold G, satisfying condition dimH¹(M,T(G)|_M)=0 all the close CR structures are induced by close embeddings.Mathematics Subject Classification (1991): 32F20, 32F10in final form: 5 September 2003     

Ribbon tableaux and the Heisenberg algebra

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: a Pieri and dual-Pieri formula for ribbon functions, a ribbon Murnaghan-Nakayama formula, ribbon Cauchy and dual Cauchy identities, and a -algebra isomorphism _n:(q)(q) which sends each to .Our study of the functions will be connected to the Fock space representation F of via a linear map :F(q) which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [KMS] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of . Identifying the Fock Space of H with the ring of symmetric functions (q) we will show that is in fact a map of H-modules with remarkable properties. The study of this map will lead to our identities concerning ribbon tableaux generating functions. We will also give a combinatorial proof that the ribbon Murnaghan-Nakayama and Pieri rules are formally equivalent.     

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Let J be an abelian surface with a generic ample line bundle . For n≥1, the moduli space M_J(2,0,2n) of (1)-semistable sheaves F of rank 2 with Chern classes c₁(F)=0, c₂(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of M_J(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of M_J(2,0,2n) for n≥2. Jaeyoo Choy was partially supported by KRF 2003-070-C00001 Young-Hoon Kiem was partially supported by a KOSEF grant R01-2003-000-11634-0.     

Compactifications of contractible affine 3-folds into smooth Fano 3-folds with <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript>=2

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space into smooth Fano 3-folds with B₂=1. In this paper, we consider the compactifications of the contractible affine 3-folds X (not necessarily X=) into smooth Fano 3-folds V with B₂=2. Consequently, we classify all such compactifications X↪(V,D₁∪D₂) in the case where K_V+D₁+D₂ is not nef. Furthermore, we see that infinitely many mutually non-isomorphic exotic 's can be compactified into Fano 3-folds with B₂=2. This phenomenon never occurs when B₂=1. During this research the author was supported as a Twenty-First Century COE Kyoto Mathematics Fellow.     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

Distribution of harmonic sums and Bernoulli polynomials modulo a prime

For a fixed integer s≥1, we estimate exponential sums with harmonic sums individually and on average, where H_s(n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H_s(n). For example, our estimates imply that for any ɛ>0, the set {H_s(n):n<p^1/2+ɛ} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as with max{n_ν|ν=1,. . . , 7}≤p^11/12+ɛ, and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values B_p−r(n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche. During the preparation of this paper, F. L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I. S. was supported in part by ARC grant DP0211459.