ω-k- torsionfree modules and ω-left approximation dimension

The notion of ω-k-torsionfree modules with respect to a bimodule ω is introduced, which is characterized in terms of left add_R ω-approximations. The notion of ω-left approximation dimension is introduced, and the forms ofk-syzygy modules beingk-torsionfree modules are described.     

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ ⁺ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ ⁺ strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ ⁺, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ ⁺ strongly compact.     

On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 2: Time-dependent theory

In part 1

1 Math. meth. in the Appl. Sci, 10, 125–144 (1988).

we studied the principle of limiting absorption for local perturbations Ω of the n-dimensional domain Ω₀ = ?^n?1 × (0, π). In this second part we extend our investigations to the time-dependent theory and show that absence of admissible standing waves implies the validity of the principle of limiting amplitude for every frequency ω≥0 if n ≠ 3 and for ω ≠ 2, 3,… if n = 3, respectively. In particular, the principle of limiting amplitude holds for every ω≥0 in the case n ≠ 3 and for every ω ≠ 2, 3,… in the case n = 3 if Ω≠Ω₀ and ν · x ′ ?0 on ?Ω, where x ′ = (x₁,…, x_n?1, 0) and ν is the normal unit vector on ?Ω pointing into the complement of Ω This result stands in remarkable contrast to the fact that both principles are violated in the case of the unperturbed domain Ω₀ at the frequencies ω = 1, 2,… if n?3. The question of the asymptotic behaviour of the solution as t→∞ for n = 3 and ω = 2, 3,… will be discussed in two subsequent papers.     

Varieties of Lattices with Geometric Descriptions

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.     

On a type of distributivity of lattice ordered groups

Let m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r _m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.     

On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 1: Time-independent theory

Let Ω be a local perturbation of the n-dimensional domain Ω₀ = Ropf;^{n ? 1} × (0, π). In a previous paper⁸ we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω₀, and v · x ′ ≤ 0 on δΩ, where x′ = ( x ₁,…, x_{n ? 1}, 0) and v is the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω₀ at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.     

Geometric quantization on presymplectic manifolds

In this paper, we discuss the possibilities of adapting geometric quantization to presymplectic manifolds, i.e., differentiable manifoldsM ^2n+k (k>0) endowed with a closed 2-form ω of rank2n. We show that such an adaptation is possible in various manners, and that, as a general idea, it reduces the quantization onM to quantization on the symplectic quotientM/V, whereV is the foliation defined by the annihilator of ω.     

Sous-espaces invariants par translations bilatérales de certains espaces de Hilbert de suites quasi-analytiquement pondérées

We use the theory of asymptotically holomorphic functions in the disc to study the lattice of left-invariant subspaces of ℓ_ω², where ω is nonincreasing, in the quasianalytic case log ω(n)/n² = +∞. When (ω(−n))_n≥0 satisfies suitable growth and regularity conditions, we show in particular that all bilaterally invariant subspaces of ℓ_ω² are generated by their intersection with ℓ_ω². When ω(n) = 1 for n ≥ 0 and ω(n) = ¦n¦/e^{1+log ¦n¦} for n < 0 this shows that all nontrivial bi-invariant subspaces of ℓ_ω² are generated by the Fourier transform of a nonconstant singular inner function.     

Countably Categorical Structures with n-Degenerate Algebraic Closure

We study the class of ω-categorical structures with n-degenerate algebraic closure for some n ε ω, which includes ω-categorical structures with distributive lattice of algebraically closed subsets (see [4]), and in particular those with degenerate (trivial) algebraic closure. We focus on the models of ω-categorical universal theories, absolutely ubiquitous structures, and ω-categorical structures generated by an indiscernible set. The assumption of n-degeneracy implies total categoricity for the first class, stability for the second, and ω-stability for the third.     

Some properties of C(X), I

By a result of A.V. Arhangel'skiǐ and E.G. Pytkeiev, the space C(X) of the continuous real functions on X with the topology of pointwise convergence has tightness ω iff Xⁿ is Lindelöf for every n ∈ ω. In this paper we describe other convergence properties of C(X) (e.g. the Fréchet-Urysohn properly) in terms of covering properties of X.In some cases the equivalence between these properties turn out to be dependent on the set theory we choose. Some open problems are also stated.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

Higher degrees of distributivity in complete generalized <Emphasis Type="Italic">MV</Emphasis>-algebras

We apply the notion of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis. Let M be a complete GMV-algebra and let α be a cardinal. We prove that M is α-distributive if and only if it is (α, 2)-distributive. We deal with direct summands of M which are homogeneous with respect to higher degrees of distributivity.     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

On the analytic continuation of Eisenstein series for Siegel's modular group of degreen

For \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) ∈ Γ(n)=Sp(n?) andZ=Z+iY,Y > 0, set $$M\left\langle Z \right\rangle = (AZ + B)(CZ + D)^{ - 1} = X_M + iY_M ;M\{ Z\} = CZ + D.$$ Denote with Γ _n (n) the subgroup defined byC=0. Forr∈? and a complex variable ω form the Eisenstein series $$E(n,r,Z,\omega ) = \sum\limits_{M\varepsilon I'_n (n)\backslash \Gamma (n)} {(DetM\{ Z\} )^{ - 2r} (DetY_M )^{\omega - r} } .$$ It is proved thatE(n, r, Z, ω) can be meromorphically continued to the ω-plane and satisfies a functional equation. Forr=1, 2, [(n?1)/2], [(n+1)/2] the functionE(n, r, Z, ω) is holomorphic at ω-r. For 3≤r≤[(n?3)/2] the functionE(n, r, Z, ω) may have poles at ω=r. But the pole-order is for two smaller than known until now. This result says especially that the Eisenstein series has Hecke summation forr=1, 2, [(n?1)/2], [(n+1)/2].     

Products without remote points

It is shown that ω × Y^ω does not have remote points if Y is a compact space with cellularity larger than ω₁. It is also shown that it is consistent that ω × Y^ω does not have remote points if Y is compact with uncountable cellularity. As an application we construct a compact space with weight ω₂ · c which can be covered by nowhere dense P-sets and a compact space with weight c for which it is independent that it can be covered by nowhere dense P-sets.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

Resonances in periodic media

We study the propagation of linear acoustic waves (a) in an infinite string with a periodic material distribution, (b) in an infinite cylinder with a meterial distribution that is periodic in the longitudinal direction and does not depend on the transverse coordinates. We assume that the wave field is generated by a time-harmonic force distribution of frequency ω acting in a compact set. We show in both cases that resonances of order t^1/2 occur for a discrete set of frequencies and that the solution is bounded as t→∞ for the remaining frequencies. In case (a) ω is a resonance frequency if and only if ω² is a boundary point of one of the spectral bands of the corresponding spatial differential operator of Hill's type. A similar characterization of the resonance frequencies is given in case (b).