The Equation x<Superscript>i</Superscript> = y<Superscript>j</Superscript>z<Superscript>k</Superscript> in a Free Semigroup

The equation xⁱ = y^jz^k has only periodic solutions in a free semigroup. This result was first proven by Lyndon and Schützenberger. We present a very short proof of this classical result. Moreover, we establish that the power of two or more of a primitive word cannot be factorized into conjugates of a different word.     

Homogenization of the parabolic cauchy problem in the Sobolev class H<Superscript>1</Superscript>(R<Superscript>d</Superscript>)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in R^d is studied. The coefficients are assumed to be periodic in R^d with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(R^d) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H ¹(R^d) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H ¹(R^d) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.     

S<Superscript>1</Superscript>-Valued Sobolev maps

We describe the structure of the space W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) , where 0 < s < ∞ and 1 ≤ p < ∞. According to the values of s, p, and n, maps in W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) can either be characterised by their phases or by a couple (singular set, phase).     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

Prevalence of Multifractal Functions in S<Superscript>v</Superscript> Spaces

Spaces called S^v were introduced by Jaffard [16] as spaces of functions characterized by the number ≃ 2^ν(α)j of their wavelet coefficients having a size ≳ 2^−αj at scale j . They are Polish vector spaces for a natural distance. In those spaces we show that multifractal functions are prevalent (an infinite-dimensional “almost-every”). Their spectrum of singularities can be computed from ν, which justifies a new multifractal formalism, not limited to concave spectra.     

Image Projection and Representation on S<Superscript>n</Superscript>

In signal processing, communications, and other branches of information technologies, it is often desirable to map the higher-dimensional signals on Sⁿ. In this article we introduce a novel method of representing signals on Sⁿ. This approach is based on geometric function theory, in particular on the theory of quasiregular mappings. The importance of sampling is underlined, and new geometric sampling theorems for general manifolds are presented.     

Pseudo-self-affine tilings in ℝ<Superscript>d</Superscript>

It is proved that every pseudo-self-affine tiling in ?^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoï tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and substitution Delone sets developed by Lagarias and Wang. Bibliography: 18 titles.     

The Katchalski-Lewis Transversal Problem in R<Superscript>n</Superscript>

Let F be a family of disjoint translates of a compact convex set in the plane. In 1980 Katchalski and Lewis showed that there exists a constant k, independent of F, such that if each three members of F are met by a line, then a "large" subfamily G ⊂ F, with |F\G| ≤ k, is met by a line. In this paper we obtain a higher-dimensional analogue containing the Katchalski-Lewis result. Also we give two constructions of families of pairwise disjoint translates of the unit ball in R³ which answer some related questions.     

Free affine actions of unipotent groups on C<Superscript>n</Superscript>

We consider free affine actions of unipotent complex algebraic groups on Cⁿ and prove that such actions admit an analytic geometric quotient if their degree is at most 2. Moreover, we classify free affine C²-actions on Cⁿ of degree n - 1 and n - 2. For every n > 4, an action of degree n - 2 appears in the classification whose quotient topology is not Hausdorff.     

Indecomposable cycles on special quartics in ℙ<Superscript>3</Superscript>

 In this note we give an example of an indecomposable higher Chow cycle on a special family of quartics in ℙ³. The example is obtained as an extension of a cycle in the higher Chow group CH ² (K,1) of a singular Kummer surface. Received: 19 June 2002 / Revised version: 17 October 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14C15, 14C25, 14H10, 14H28, 14F42     

On the four-dimensional T<Superscript>2</Superscript>-manifolds of positive Ricci curvature

We prove that there is a T ²-invariant Riemannian metric of positive Ricci curvature on every four-dimensional simply connected T ²-manifold.     

Generic Immersions of the Two-Sphere to R<Superscript>3</Superscript> and Their Skeletons

Let f:S ² ↬ R ³ be a generic smooth immersion. The skeleton of f is the following triple (Γ, D, p): Γ ⊂ R ³ is the 1-polyhedron of singular points of f, D = f⁻¹(Γ) ⊂ S ² is also a 1-polyhedron, and p : D → Γ, x ↦ f(x), is the projection. For triples of the form (D,Γ,p), where Γ has at most four vertices, we give an iff-condition under which the triple is the skeleton of a smooth immersion f : S ² ↬ R ³. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 300–313.     

Minty Variational Inequalities,Increase-Along-Rays Property and Optimization<Superscript>1</Superscript>

Let E be a linear space, let K E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^,K ), which can be considered as a generalization of MVI (f ^,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^K ) and the property of f to increase-along-rays starting at x ^*, fIAR (K,x ^*). We prove that the GMVI (f ^,K ) with radially l.s.c. function f has a solution x ^* ker K if and only if fIAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^,K ) has a solution x ^*K, then x ^* is a global minimizer of the problem min f(x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.     

A Converse Hawking-Unruh Effect and dS<Superscript>2</Superscript>/CFT Correspondence

Given a local quantum field theory net $ \mathcal{A} $ on the de Sitter spacetime dS ^d , where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e., particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS ^d . Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets $ \mathcal{A} $ on dS ² and local conformal non-isotonic families (pseudonets) $ \mathcal{B} $ on S ¹. The pseudonet $ \mathcal{B} $ gives rise to two local conformal nets $ \mathcal{B}_\pm $ on S ¹, that correspond to the $ \frak{H}_\pm $ horizon components of $ \mathcal{A} $, and to the chiral components of the maximal conformal subnet of $ \mathcal{A} $. In particular, $ \mathcal{A} $ is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on $ \frak{H}_\pm $ have positive energy and the translations on $ \frak{H}_\mp $ are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS ² has positive/negative generator. Communicated by Klaus Fredenhagen submitted 07/02/03, accepted: 07/07/03     

Topological hypercovers and <Superscript>1</Superscript>-realizations

We show that if U_* is a hypercover of a topological space X then the natural map hocolim U_* X is a weak equivalence. This fact is used to construct topological realization functors for the ¹-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.Mathematics Subject Classification (2000):55U35, 14F20, 14F42The second author was supported by an NSF Postdoctoral Research Fellowship     

Hyperbolic linear Weingarten surfaces in ℝ<Superscript>3</Superscript>

A hyperbolic linear Weingarten surface in ℝ³ is a surface M whose mean and Gaussian curvatures satisfy the relationship 2aH +bK = c for real numbers a, b, c such that a²+bc < 0. In this work we obtain a representation for such a surface in terms of its Gauss map when, more generally, a, b, c are functions on M. We also study the completeness of such surfaces and describe a procedure to construct complete examples from solutions of the sine-Gordon equation. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant no. PAI-05-034. The first and second authors are partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746.     

Fractional integration associated with second order divergence operators on R<Superscript>n</Superscript>

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (−Δ)^−α/2 are extended to the generalised fractional integrals L ^–α/2 for 0 < α < n, where L=−div A∇ is a uniformly complex elliptic operator with bounded measurable coefficients in ℝⁿ.     

Double bubbles in<Emphasis Type="Bold">S</Emphasis><Superscript>3</Superscript> and<Emphasis Type="Bold">H</Emphasis><Superscript>3</Superscript>

We prove the double bubble conjecture in the three-sphereS ³ and hyperbolic three-spaceH ³ in the cases where we can apply Hutchings theory:

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.