Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M ⁿ suitable uniform structures, obtain completed spaces ^b,m M or M ^r (I, B _k ), respectively and calculate for each component of M ^r (I, B _k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

On the condensation points of the lagrange spectrum

For a positive integerN, L(N) denotes the set of Lagrange values of all sequences (a _k:k=0, ±1, ±2,…) of positive integers with lim sup _k ^ak=N. It is shown that for anyN≥3L(N) has infinitely many condensation points. Such points can be realized as Markov values of symmetric doubly periodic sequences whose period consists of a semi-symmetric tuple.     

Properties of Integers and Finiteness Conditions for Semigroups

Let h and k be integers greater than 1; we prove that the following statements are equivalent: 1) the direct product of h copies of the additive semigroup of non-negative integers is not k-repetitive; 2) if the direct product of h finitely generated semigroups is k-repetitive, then one of them is finite. Using this and some results of Dekking and Pleasants on infinite words, we prove that certain repetitivity properties are finiteness conditions for finitely generated semigroups.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Absolutely continuous cocycles over irrational rotations

For homeomorphisms $$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$ (z, w ∈S ¹,α is irrational,?:S ¹→S ¹) of the torusS ¹×S ¹ it is proved thatT? has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions whenever? is absolutely continuous with nonzero topological degree and the derivative of? is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

The harmonic Bergman kernels for punctured domains

We obtain the explicit formulae for the harmonic Bergman kernels of Bn／{0} and Rn／Bn and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel.We also get the explicit formula for the weighted harmonic Bergman kernel of Bn／{0} with the weight 1/｜x｜4.     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

Let K be a Calderón–Zygmund convolution kernel on ?. We discuss the L ^p -boundedness of the maximal directional singular integral $$T_{\mathbf{V}} f (x)= \sup_{v \in \mathbf{V}} \bigg| \int_{\mathbb{R}} f(x+t v) K(t) \, \mathrm{d} {t} \bigg| $$ where V is a finite set of N directions. Logarithmic bounds (for 2≤p<∞) are established for a set V of arbitrary structure. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L ^p almost orthogonality principle for Fourier restrictions to cones.     

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

For a positive integer k, let k?+?k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k?≥?1, there exists a constant c _k so that First Fit will use at most $c_kw^2$ chains in partitioning a poset P of width at most w, provided the poset excludes k?+?k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w ^16logw ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead’s exponential bound: (5 ^w ???1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek–Krawczyk–Szczypka bound for the performance of First Fit to 8(k???1)² w, which in turn yields the modest improvement to O(w ^14logw ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t?=?t(k,w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k?+?k as a subposet.     

A characterization of Poissonian domains

We give a characterization of Possonian domains inR ⁿ , i.e., those domains for which every bounded harmonic function is the harmonic extension of some function inL ^∞ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.     

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

The non-gap sequence of a subcode of a generalized Reed–Solomon code

This paper addresses the question how often the square code of an arbitrary l-dimensional subcode of the code GRS _k (a, b) is exactly the code GRS_2k-1(a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first used and stated by Wieschebrink where he applied the Sidelnikov–Shestakov attack to break the Berger–Loidreau cryptosystem.     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.