Syzygies of grids of projective lines

In this paper we study the syzygy modules of a grid or a fat grid of . We compute the minimal free resolution for the ideal of a complete grid in , and we conjecture this resolution in . Moreover we compute the minimal free resolution for the ideal of an incomplete grid of . We also conjecture the minimal free resolution for the ideal of a fat complete grid in .      

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

Given an inclusion of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of implies that of the scaling limit of . As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion of local nets with the same canonical field net , we find sufficient conditions which entail the equality of the canonical field nets of . Work supported by MIUR, GNAMPA-INDAM, the EU and SNS. Submitted: August 29, 2008. Accepted: March 23, 2009.     

Lagrangian minimal surfaces in Lorentzian complex plane

Flat Lagrangian minimal surfaces in the Lorentzian complex plane are classified by B. Y. Chen and L. Vrancken in [8]. On the other hand, Vrancken proves in [11] that Lagrangian minimal surfaces of constant curvature in are flat surfaces. In this article, we classify all Lagrangian minimal surfaces in which are free from flat points. Received: 16 December 2007     

Scales for co-compact embeddings of virtually free groups

Let Γ be a group which is virtually free of rank at least 2 and let be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family .      

Multivariable moment problems

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C^*-algebras generated by universal row contractions associated with , the free semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra (resp. ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as ,and, more generally, the quotient algebra , where J is an arbitrary two-sided ideal of . All these results are extended to the generalized Cuntz algebra , where G_i⁺ are the positive cones ofdiscrete subgroups G_i⁺ of the real line . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra , where J is an arbitrary two-sided ideal ofthe free semigroup algebra .     

The norm of products of free random variables

Let X _i denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X _i it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX _i = 1, then is between and c ₂ n for certain constant c ₁ and c ₂. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X _i , and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables.     

Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

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.     

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

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 .     

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.      

Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F _n inside mapping class groups, Aut(), and Out() which have infinitely many conjugacy classes of finite-order elements.      

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.     

Chains and antichains in partial orderings

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is or , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two sets. Our main result is that there is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering has the feature that for every , there is an infinite chain or antichain that is relative to , then we have uniform dichotomy: either for all copies of , there is an infinite chain that is relative to , or for all copies of , there is an infinite antichain that is relative to .     

Anisotropic version of a theorem of H. Hopf

Given a positive function F on S ² which satisfies a convexity condition, we define a function for surfaces in which is a generalization of the usual mean curvature function. We prove that an immersed topological sphere in with = constant is the Wulff shape, up to translations and homotheties.      

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

Maximum decay rate for the nonlinear Schrödinger equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation in with and We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space in the energy space, namely or in according to the different cases.