$\mathcal{C}^{2}$ surface diffeomorphisms have symbolic extensions

We prove that C²\mathcal{C}^{2} surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. Following the strategy of Downarowicz and Maass (Invent. Math. 176:617–636, 2009) we bound the local entropy of ergodic measures in terms of Lyapunov exponents. This is done by reparametrizing Bowen balls by contracting maps in a approach combining hyperbolic theory and Yomdin’s theory.     

$${\mathcal{C}}_{0}$$ ( 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     

Tight convex underestimators for $${\mathcal{C}^2}$$ -continuous problems: II. multivariate functions

In Part I (Gounaris, C.E., Floudas, C.A.: Tight convex understimators for -continuous functions: I: Univariate functions. J. Global Optim. (2008). doi: ), we introduced a novel approach for the underestimation of univariate functions which was based on a piecewise application of the well-known αBB underestimator. The resulting underestimators were shown to be very tight and, in fact, can be driven to coincide with the convex envelopes themselves. An approximation by valid linear supports, resulting in piecewise linear underestimators was also presented. In this paper, we demonstrate how one can make use of the high quality results of the approach in the univariate case so as to extend its applicability to functions with a higher number of variables. This is achieved by proper projections of the multivariate αBB underestimators into select two-dimensional planes. Furthermore, since our method utilizes projections into lower-dimensional spaces, we explore ways to recover some of the information lost in this process. In particular, we apply our method after having transformed the original problem in an orthonormal fashion. This leads to the construction of even tighter underestimators, through the accumulation of additional valid linear cuts in the relaxation.     

A class of {\Sigma _{3}^{0}} modular lattices embeddable as principal filters in {\mathcal{L}^{\ast }(V_{\infty })}

Let I ₀ be a a computable basis of the fully effective vector space V _∞ over the computable field F. Let I be a quasimaximal subset of I ₀ that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter ${\mathcal{L}^{\ast}(V,\uparrow )}$ of V = cl(I) is isomorphic to the lattice ${\mathcal{L}(n, \overline{F})}$ of subspaces of an n-dimensional space over ${\overline{F}}$ , a ${\Sigma _{3}^{0}}$ extension of F. As a corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices ${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$ is isomorphic to a principal filter of ${\mathcal{ L}^{\ast}(V_{\infty})}$ . We thus answer Question 5.3 “What are the principal filters of ${\mathcal{L}^{\ast}(V_{\infty}) ?}$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets.     

The intersection map of subgroups and the classes {\mathcal {PST}_c}, {\mathcal {PT}_c} and {\mathcal {T}_c}

A group G is called a ${\mathcal {T}_{c}}$ -group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes ${\mathcal {PT}_{c}}$ and ${\mathcal {PST}_{c}}$ are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that ${X=Y \cap H}$ . We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of ${\mathcal {T}_c}$ , ${\mathcal {PT}_c}$ and ${\mathcal {PST}_c}$ groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Approximation of solution operators of elliptic partial differential equations by {\mathcal{H}}- and {\mathcal{H}^2}-matrices

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse ${\mathcal{H}}$ - and ${\mathcal{H}^2}$ -matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for ${\mathcal{H}}$ - and ${\mathcal{H}^2}$ -matrix approximations of the entire matrices.     

Proper trajectories of type {\mathbb{C}^{\ast}} of a polynomial vector field on {\mathbb{C}^2}

We prove that if a polynomial vector field on ${\mathbb{C}^2}$ has a proper and non-algebraic trajectory analytically isomorphic to ${\mathbb{C}^{\ast}}$ all its trajectories are proper, and except at most one which is contained in an algebraic curve of type ${\mathbb{C}}$ all of them are of type ${\mathbb{C}^{\ast}}$ . As corollary we obtain an analytic version of Lin?CZa?denberg Theorem for polynomial foliations.     

Maximally tangent complex curves for germs of finite type {\mathcal{C}^\infty} pseudoconvex domains in {\mathbb C^3}

In this paper we discuss D’Angelo finite type pseudoconvex domains Ω in ${\mathbb C^3}$ . We are interested in complex curves tangent to higher order. Our main result is that there are only finitely many curves of maximal type. Maximal type has to be taken in a micro-local sense since the maximal type can be different in different directions. And of course to get finiteness we have to ignore higher order irrelevant terms which can be added without restriction. In the process of describing such a curve we find a singular change of coordinates which reduces the curve to a complex line.     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Finite determinacy and Whitney equisingularity of map germs from {\mathbb{C}^n} to {\mathbb{C}^{2n-1}}

We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D ²(f)) = 2μ(D ²(f)/S ₂)+C(f)?1, where D ²(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f _t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f _t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.     

On Holomorphic Automorphisms of a Class of Non-homogeneous Rigid Hypersurfaces in C^N＋1

The author determines the real-analytic infinitesimal CR automorphisms of a class of non-homogeneous rigid hypersurfaces in C^N＋1 near the origin, and the connected component containing the identity transformation of all locally holomorphic automorphisms of these hypersurfaces near the origin.     

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\) , that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\) -orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\) .     

$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.     

Amenability,tubularity, and embeddings into $$\mathcal{R}^{\omega}$$

Suppose M is a tracial von Neumann algebra embeddable into (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε >  0 there exist and γ > 0 such that if then there exists a k × k unitary u satisfying for each 1 ≤  i ≤  n. We show that the following conditions are equivalent:

High-Order \mathcal{D}^{\alpha}-Type Iterative Learning Control for Fractional-Order Nonlinear Time-Delay Systems

This paper presents a high-order $\mathcal{D}^{\alpha}$ -type iterative learning control (ILC) scheme for a class of fractional-order nonlinear time-delay systems. First, a discrete system for $\mathcal{D}^{\alpha}$ -type ILC is established by analyzing the control and learning processes, and the ILC design problem is then converted to a stabilization problem for this discrete system. Next, by introducing a suitable norm and using a generalized Gronwall–Bellman Lemma, the sufficiency condition for the robust convergence with respect to the bounded external disturbance of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.