Definability in substructure orderings, IV: Finite lattices

Let ${\mathcal{L}}$ be the ordered set of isomorphism types of finite lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. Our main result is that for every finite lattice L, the set {?, ? ^opp} is definable, where ? and ? ^opp are the isomorphism types of L and its opposite (L turned upside down). We shall show that the only non-identity automorphism of ${\mathcal{L}}$ is the map ${\ell \mapsto \ell^{\rm opp}}$ .     

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Degrees of isomorphism types and countably categorical groups

It is shown that for every Turing degree d there is an ω-categorical group G such that the isomorphism type of G is of degree d. We also find an ω-categorical group G such that the isomorphism type of G has no degree.     

Unit Balls, Lorentz Boosts, and Hyperbolic Geometry

The bounded symmetric spaces naturally associated with the Poincaré and Beltrami-Klein models of hyperbolic geometry on the open unit ball B in ${\mathbb{R}^n}$ and with the automorphism group of biholomorphic maps on the open ball in ${\mathbb{C}^n}$ give rise by a standard construction to specialized loop structures (nonassociative groups), which we use to define canonical metrics, called rapidity metrics. We show that this rapidity metric agrees with the classical Poincaré metric resp. the Cayley-Klein metric resp. the Bergman metric. We introduce the Lorentz boost of vectors in B, which turns out to be a loop isomorphism. It induces a similarity of metrics between the rapidity metric of the Einstein or Möbius loop and the trace metric on positive definite matrices restricted to the Lorentz boosts.     

Cross-Ratios Over Local Alternative Rings

It is shown that the conjugacy relation is an equivalence relation on the alternative field A and on the local alternative ring R = A + Aε of dual numbers over A, but on no other alternative ring of a certain class. On the projective lines over A and R the cross-ratio of four points is defined as a conjugacy class. Its elementary properties and applications to chain geometries are investigated.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

Definibility in normal theories

This paper initiates an investigation which seeks to explain elementary definability as the classical results of mathematicallogic (the completeness, compactness and Löwenheim-Skolem theorems) explain elementary logical consequence. The theorems of Beth and Svenonius are basic in this approach and introduce automorphism groups as a means of studying these problems. It is shown that for a complete theoryT, the definability relation of Beth (or Svenonius) yields an upper semi-lattice whose elements (concepts) are interdefinable formulas ofT (formulas having equal automorphism groups in all models ofT). It is shown that there are countable modelsA ofT such that two formulae are distinct (not interdefinable) inT if and only if they are distinct (have different automorphism groups) inA. The notion of a concepth being normal in a theoryT is introduced. Here the upper semi-lattice of all concepts which defineh is proved to be a finite lattice—anti-isomorphic to the lattice of subgroups of the corresponding automorphism group. Connections with the Galois theory of fields are discussed.     

Stable structures homogeneous for a finite binary language

LetL be a finite relational language andH(L) denote the class of all countable stableL-structuresM for which Th(M) admits elimination of quantifiers. ForM ∈H(L) define the rank ofM to be the maximum value of CR(p, 2), wherep is a complete 1-type over Ø and CR(p, 2) is Shelah’s complete rank. IfL has only unary and binary relation symbols there is a uniform finite bound for the rank ofM ∈H(L). This theorem confirms part of a conjecture of the first author. Intuitively it says that for eachL there is a finite bound on the complexity of the structures inH(L).     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Definability in the Substructure Ordering of Simple Graphs

For simple graphs, we investigate and seek to characterize the properties first-order definable by the induced subgraph relation. Let \({\mathcal{P}\mathcal{G}}\) denote the set of finite isomorphism types of simple graphs ordered by the induced subgraph relation. We prove this poset has only one non-identity automorphism co, and for each finite isomorphism type G, the set {G, G ^co } is definable. Furthermore, we show first-order definability in \({\mathcal{P}\mathcal{G}}\) captures, up to isomorphism, full second-order satisfiability among finite simple graphs. These results can be utilized to explore first-order definability in the closely associated lattice of universal classes. We show that for simple graphs, the lattice of universal classes has only one non-trivial automorphism, the set of finitely generated and finitely axiomatizable universal classes are separately definable, and each such universal subclass is definable up to the unique non-trivial automorphism.     

Padé tables of entire functions of very slow and smooth growth

Letf(z)=σ _j?o ^∞ a _j z ^j be entire with $$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$ whereρ ₀=0.4559... is the positive root of the equation $$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$ . It is shown that the Padé table off is normal, and asL→∞, [L/M _L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M _L } _L=1 ^∞. In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ ₀ is shown to be best possible in a strong sense.     

A test for identities satisfied in lattices of submodules

SupposeR is ring with 1, andH?(R) denotes the variety of modular lattices generated by the class of lattices of submodules of allR-modules. An algorithm using Mal'cev conditions is given for constructing integersm≧0 andn≧1 from any given lattice polynomial inclusion formulad∈e. The main result is thatd∈e is satisfied in every lattice inH?(R) if and only if there existsx inR such that (m·1)x=n·1 inR, where 0·1=0 andk·1=1+1...+1 (k times) fork≧1. For example, this “divisibility” condition holds form=2 andn=1 if and only if 1+1 is an invertible element ofR, and it holds form=0 andn=12 if and only if the characteristic ofR divides 12. This result leads to a complete classification of the lattice varietiesH?(R),R a ring with 1. A set of representative rings is constructed, such that for each ringR there is a unique representative ringS satisfyingH?(R)=H?(R). There is exactly one representative ring with characteristick for eachk≧1, and there are continuously many representative rings with characteristic zero. IfR has nonzero characteristic, then all free lattices inH?(R) have recursively solvable word problems. A necessary and sufficient condition onR is given for all free lattices inH?(R) to have recursively solvable word problems, ifR is a ring with characteristic zero. All lattice varieties of the formH?(R) are self-dual. A varietyH?(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varietiesH?(R) is constructed.     

Separability of wild automorphisms of a polynomial ring

For a large class (including the Nagata automorphism) of wild automorphisms F of k[x, y, z] (where k is a field of characteristic zero), we prove that we can find a weight w such that there exists no tame automorphism with the same w-weight multidegree.     

Norm convergence of normalized iterates and the growth of Kœnigs maps

Let ? be an analytic function defined on the unit diskD, with ?(D)?D, ?(0)=0, and ?′(0)=λ≠0. Then by a classical result of G. K?nigs, the sequence of normalized iterates Φ _n /λ ⁿ converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH _p , the sequence Φ _n /λ ⁿ converges to σ in the norm ofH _p . We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ? is univalent. When ? is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyH _p . It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceL _a ^p .     

Embeddings into power series rings

Let T be an ordered ring without divisors of zero, and letA be the set of archimedean subgroups of T generated by a Banaschewski functionτ. Let_XΠ_Δ R be the power series ring of the real numbers ? over the totally ordered semigroup Δ of archimedean classes of T, and letχ be the usual Banaschewski function on_XΠ_Δ R. The following are equivalent:

1. τ satisfies the additional condition; for convex subgroups P,Q of T, where
2. There exists a one-to-one homomorphism Γ:T→_XΠ_Δ R of ordered rings such that for every convex subgroup Q of_XΠ_Δ R, there exists a convex subgroup P of T such that \(\Gamma (P) \subseteq Q\) and \(\Gamma (\tau (P)) \subseteq \chi (Q)\) .

     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Analysis of Basis Pursuit via Capacity Sets

Finding the sparsest solution α for an under-determined linear system of equations D α=s is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of α that allow its recovery using ? ₁-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of D called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given D, called the capacity sets. We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of D α=s.