Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H^s(ℝ²). For critical case (α=), existence of global (large) solutions in H^s is proved for s≥ when is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to is somewhat surprising and unexpected. For super-critical case (α ∈ (0,)), existence and uniqueness of global (large) solution in H^s is proved when the product is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].     

On topological Kadec norms

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let be two separable metric topologies on the same set X. We prove that every point in X has an -neighbourhood basis consisting of sets that are -closed if and only if there exists a function φ: X→ℝ that is -lower semi-continuous and such that is the weakest topology on X that contains and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.     

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain Ω in is an obstruction to compactness of the -Neumann operator on Ω, provided that at some point of M, the Levi form of bΩ has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. Research supported in part by NSF grant number DMS-0100517.     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

The Optimal Order for the p-th Moment of Sums of Independent Random Variables with Respect to Symmetric Norms and Related Combinatorial Estimates

For n independent random variables f₁, . . . ,f_n and a symmetric norm || ||_X on ℝⁿ, we show that for 1≤ p < ∞ Here is the disjoint sum of the f_i's and h* is the non-increasing rearrangement. Similar results (where L_p is replaced by a more general rearrangement invariant function space) were obtained first by Litvak, Gordon, Schütt and Werner for Orlicz spaces X and independently by S. Montgomery-Smith [22] for general X but without an explicit analysis of the order of growth for the constant in the upper estimate. The order is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms l_{f, q} on ℝⁿ under mild assumptions on f. We give applications to the theory of noncommutative L_p spaces.     

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

We prove that a complete embedded maximal surface in = (³, dx₁² + dx₂²-dx₃²) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x₃=0} in with n+12 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x₃=0}. Moreover, the position of the singular points in ³ and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space of marked graphs with n+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n–1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .Mathematics Subject Classification (2000): 53C50, 58D10, 53C42First and second authors research partially supported by MEC-FEDER grant number MTM2004-00160Second and third authors research partially supported by Consejería de Educación y Ciencia de la Junta de Andalucía and the European Union.     

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.     

Global Euler obstruction and polar invariants

Let be a purely dimensional, complex algebraic singular space. We define a global Euler obstruction Eu(Y) which extends the Euler-Poincaré characteristic in case of a nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of global polar invariants. Partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

On the exact Hausdorff dimension of the set of Liouville numbers. II

Let denote the set of Liouville numbers. For a dimension function h, we write for the h-dimensional Hausdorff measure of . In previous work, the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function increases faster than any power function near 0, then , and if h is a dimension function for which the function increases slower than some power function near 0, then . This provides a complete characterization of all Hausdorff measures of without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if then does not have σ-finite measure. This answers another question asked by R. D. Mauldin. This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University.     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

On the zero-distribution of Epstein zeta-functions

We prove that the mean value of the real parts of the nontrivial zeros of the Epstein zeta-function associated with a positive definite quadratic form in n variables is equal to . Furthermore, we show that Epstein zeta-functions in general have an asymmetric zero-distribution with respect to the critical line Re .     

Density results for the <Emphasis Type="Italic">W</Emphasis><Superscript>1/2</Superscript> energy of maps into a manifold

We show that maps from Bⁿ to a smooth compact boundaryless manifold which are smooth out of a singular set of dimension n−2 are dense for the strong topology in W^1/2(Bⁿ,). We also prove that for n≥2 smooth maps from Bⁿ to are dense in W^1/2(Bⁿ,) if and only if π₁()=0, i.e. the first homotopy group of is trivial.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

A Springer Theorem for Hermitian forms

Throughout this paper, we let (D,σ) be a central F -division algebra with involution σ such that F^σ={d ∈ F|σ(d)=d} is a Henselian valued field. By [11], the valuation on F^σ extends uniquely to a valuation on D. We denote this valuation by v. Moreover, we assume that the characteristic of the residue field, , is not 2. If the valuation on F is discrete, then any quadratic form q can be written as q= q₁⊥πq₂, where π is a uniformizer and q_i are unit forms. Springer's Theorem states that q is isotropic if and only if at least one of the residue forms and is isotropic. In this paper we generalize this result to ɛ -Hermitian forms. In Section 4, we use the connection between involutions on algebras and ɛ-Hermitian forms to prove an analog of the Springer Theorem for involutions. This paper was part of the author's doctoral dissertation at New Mexico State University. The author wishes to thank his advisor Pat Morandi for his tireless help.     

C k -estimates for the -equation on convex domains of finite type

For a bounded convex domain with C^∞ smooth boundary of finite type m and q=1, . . . ,n−1, we construct a -solving integral operator T*_q such that for all k ∈ ℕ and the usual C^k and -norms the operator is continuous.     

Characterisation of homogeneous polynomials which are constant on complex-tangential curves in the boundary of the unit ball of

We prove that a complex-tangential curve γ in the boundary of the unit ball of having the property that there exists a homogeneous polynomial P such that P=1 on γ has constant curvature. This implies that a homogeneous polynomial P having the property that there exists a closed complex-tangential curve γ (respectively a totally real 2-dimensional submanifold) in the boundary of the unit ball of such that P=1 on γ (respectively |P|=1 on γ) reduces to a monomial by a unitary chage of variables. These results represent a positive answer to conjectures of H. O. Kim.     

Beurling primes with large oscillation

A Beurling generalized number system is constructed having integer counting function N_B(x) = κx +O(x^θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π_B(x) =li(x) + Ω(xexp(-c)), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability. The author was supported in part by NSF grants DMS-0070720 and DMS-0244660. The author was supported in part by NSF grant DMS-0244660.     

Three dimensional divisorial extremal neighborhoods

In this paper we study divisorial extremal neighborhoods such that 0 ∈ X is a cA_n type threefold terminal singularity, and Γ=f(E) is a smooth curve, where E is the f-exceptional divisor. We view a divisorial extremal neighborhood as a one parameter smoothing of certain surface singularities, and based on this we give a classification of such neighborhoods.