Homology of planar polygon spaces

In this paper, we study topology of the variety of closed planar n-gons with given side lengths . The moduli space where , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces as functions of the length vector . We also find sharp upper bounds on the sum of Betti numbers of depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.      

Algebraic cycles on products of elliptic curves over <Emphasis Type="Italic">p</Emphasis>-adic fields

We show that for any odd prime p there is a smooth projective threefold W defined over a p-adic field such that the Chow group CH²(W)/ and the Griffiths group Griff²(W)/ are infinite for suitable primes . We further give examples of smooth projective fourfolds over these p-adic fields for which the -torsion subgroup CH³ is infinite.     

Derivative-free characterizations of bounded composition operators between Lipschitz spaces

We prove that maps into if and only if belongs to . In the case β < 1, we give another two equivalent conditions. Supported by MNZŽS Serbia, Project No. ON144010.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Quantization for a fourth order equation with critical exponential growth

For concentrating solutions weakly in H ²(Ω) to the equation on a domain with Navier boundary conditions the concentration energy is shown to be strictly quantized in multiples of the number .     

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Inequalities of the Brunn–Minkowski type for Gaussian measures

Let be an integer, let γ be the standard Gaussian measure on , and let . Given this paper gives a necessary and sufficient condition such that the inequality is true for all Borel sets A ₁,...,A _m in of strictly positive γ-measure or all convex Borel sets A ₁,...,A _m in of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type for γ which are true for all convex sets but not for all measurable sets.      

Bounded universal functions in one and several complex variables

We show how to obtain functions that are universal for the ball of , where . The existence of our functions will follow from universality criteria, but we also show how to construct them. Then we study the connection between certain interpolating sequences, runaway automorphisms, and the existence of universal functions on domains in .      

Inflectional loci of scrolls

Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.      

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as of solutions to a linear, parabolic system of equations with time-dependent coefficients in , where is a bounded domain. On we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function . This includes in particular situations when the coefficients may take different values on different parts of and the boundaries between them can move with t but stabilize as . The main result is an asymptotic representation of solutions for large t. A consequence is that for , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.     

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

Positivity of the spherically averaged atomic one-electron density

We investigate the positivity of the spherically averaged atomic one-electron density . For a which stems from a physical ground state we prove that for r ≥ 0. This article may be reproduced in its entirety for non-commercial purposes.     

A spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data

In this paper we investigate vector-valued parabolic initial boundary value problems , subject to general boundary conditions in domains G in with compact C ^2m -boundary. The top-order coefficients of are assumed to be continuous. We characterize optimal L ^p -L ^q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on and the Lopatinskii–Shapiro condition on are necessary for these L ^p -L ^q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.      

Calderón–Zygmund estimates for higher order systems with p(x) growth

For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0).