Recasting the Elliott conjecture

Let A be a simple, unital, finite, and exact C^*-algebra which absorbs the Jiang–Su algebra tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases— -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of -stable C^*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that -theoretic invariants will classify separable and nuclear C^*-algebras—with the recent appearance of counterexamples to its strongest concrete form. Research supported by the DGI MEC-FEDER through Project MTM2005-00934, and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. A. S. Toms was also supported in part by an NSERC Discovery Grant.     

Potential theory of truncated stable processes

For any , a truncated symmetric α-stable process is a symmetric Lévy process in with a Lévy density given by for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails. The research of Panki Kim is supported by Research Settlement Fund for the new faculty of Seoul National University. The research of Renming Song is supported in part by a joint US-Croatia grant INT 0302167.     

Links of symbolic powers of prime ideals

In this paper, we prove the following. Let be a Cohen-Macaulay local ring of dimension d ≥ 2. Suppose that either R is not regular or if R is regular assume that d ≥ 3. Let t ≥ 2 be a positive integer. If is a regular sequence contained in , then

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B.Gross. This conjecture predicts some -adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when is an Archimedean place. Here we consider the conjecture when is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.     

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

Interfaces for 1-D degenerate Keller–Segel systems

Let us consider the Keller–Segel system of degenerate type (KS) _m with m >1 below. We prove the property of finite speed of propagation for weak solutions u with a certain regularity. Moreover, we investigate the interface curve which separates the regions and . Concretely, we characterize the interface curve as the solution of a certain ordinary differential equation associated with (KS) _m .     

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 .      

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation $\bar{F}$ is a difference quotient, that is where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances. Mathematics Subject Classification (2000) 47J20, 58B20, 49J45     

Stable embedded minimal surfaces bounded by a straight line

We prove that if is a properly embedded oriented stable minimal surface whose boundary is a straight line and the area of M in extrinsic balls grows quadratically in the radius, then M is a half-plane or half of the classical Enneper minimal surface. This solves a conjecture posed by B. White in Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, International Press, Somerville, 1996.     

Differentiable Functionals and Smoothed Bootstrap

The differentiability properties of statistical functionals have several interesting applications. We are concerned with two of them. First, we prove a result on asymptotic validity for the so-called smoothed bootstrap (where the artificial samples are drawn from a density estimator instead of being resampled from the original data). Our result can be considered as a smoothed analog of that obtained by Parr (1985, Stat. Probab. Lett., 3, 97-100) for the standard, unsmoothed bootstrap. Second, we establish a result on asymptotic normality for estimators of type generated by a density functional being a density estimator. As an application, a quick and easy proof of the asymptotic normality of , (the plug-in estimator of the integrated squared density ) is given.     

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.     

The Automorphism Conjecture for Small Sets and Series Parallel Sets

The automorphism conjecture for ordered sets states that the automorphism to endomorphism ratio will tend to zero as the size of the ordered set goes to infinity. We show by computer enumeration that up to size 11 the ratio is largest for weakly ordered sets. Subsequently, we derive exact recursive formulas for the number of homomorphisms between two related types of weakly ordered sets and we prove a strong automorphism conjecture for series-parallel ordered sets. We conclude with an example that shows that the automorphism to endomorphism ratio can exceed for arbitrarily large .This work was sponsored by Louisiana Board of Regents RCS grant LEQSF(1999-02)-RD-A-27.     

LDPC codes from the Hermitian curve

In this paper, we study the code which has as parity check matrix the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that has a double cyclic structure and that by shortening in a suitable way it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix via a geometric approach.      

Self-Reciprocal Irreducible Pentanomials Over \mathbb{F}_2

Joseph Yucas and Gary Mullen conjectured that there is no self-reciprocal irreducible pentanomial of degree n over if n is divisible by 6. In this note we prove this conjecture for the case n ≡ 0, and disprove the conjecture for the case n ≡ 6 (mod 12) AMS Classifications: 11T55     

Rational invariants of certain classical groups over finite fields

Let be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over and be the rational function field over . We seek to understand the structure of the rational invariant subfield . In this paper, we prove that is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.      

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

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).     

A Marstrand theorem for measures with polytope density

Given and any centrally symmetric convex polytope , define we prove that if a Radon measure μ has the property then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in (Marstrand in Trans Am Math Soc 205:369–392, 1964).     

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of , let denote the field of rational functions such that . We will say that F is quasi-locally finite if there exists a nonzero such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each the sequence is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.