Countable Sequence Research Articles

Semilattices of Groups and Nonstable K-Theory of Extended Cuntz Limits

We give an elementary characterization of those abelian monoids M that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product � × G, where � is a distributive semilattice with zero and G is an abelian group. When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with K.R. Goodearl.

Easiness in graph models

We generalize Baeten and Boerboom's method of forcing to show that there is a fixed sequence ( u k ) k ∈ ω of closed (untyped) λ -terms satisfying the following properties: (a) For any countable sequence ( g k ) k ∈ ω of Scott continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that ( λ x . xx ) ( λ x . xx ) u k represents g k in the model. (b) For any countable sequence ( t k ) k ∈ ω of closed λ -terms there is a graph model that satisfies ( λ x . xx ) ( λ x . xx ) u k = t k for all k . We apply these two results, which are corollaries of a unique theorem, to prove the existence of (1) a finitely axiomatized λ -theory L such that the interval lattice constituted by the λ -theories extending L is distributive; (2) a continuum of pairwise inconsistent graph theories (= λ -theories that can be realized as theories of graph models); (3) a congruence distributive variety of combinatory algebras (lambda abstraction algebras, respectively).

SAMPLING SEQUENCES OF COMPACTLY SUPPORTED DISTRIBUTIONS IN Lp(R)

The aim of the paper is to obtain some generalizations of the so-called Plancherel–Polya inequalities which are also known as frame inequalities. By using these inequalities we show that a function f ∈ Lp(R), 1 ≤ p ≤ ∞, which is entire function of exponential type is uniquely determined by a set of numbers {Φj(f)}, j ∈ ℕ where {Φj}, j ∈ ℕ is a countable sequence of compactly supported distributions. In the case p = 2 we offer two reconstruction methods of a function f from a sequence of samples {Φj(f)}, j ∈ ℕ. The first reconstruction algorithm is given in terms of frames. To describe our second algorithm we introduce the so-called average variational splines.

Specker sequences revisited

AbstractSpecker sequences are constructive, increasing, bounded sequences of rationals that do not converge to any constructive real. A sequence is said to be a strong Specker sequence if it is Specker and eventually bounded away from every constructive real. Within Bishop's constructive mathematics we investigate non‐decreasing, bounded sequences of rationals that eventually avoid sets that are unions of (countable) sequences of intervals with rational endpoints. This yields surprisingly straightforward proofs of certain basic results fromconstructive mathematics. Within Russian constructivism, we show how to use this general method to generate Specker sequences. Furthermore, we show that any nonvoid subset of the constructive reals that has no isolated points contains a strictly increasing sequence that is eventually bounded away from every constructive real. If every neighborhood of every point in the subset contains a rational number different from that point, the subset contains a strong Specker sequence. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The positive foundation of the common prior assumption

The existence of a common prior is a property of the state space used to model the players' incomplete information. We show that this property is not just a technical artifact of the model, but that it is immanent to the players' beliefs. To this end, we devise a condition, phrased solely in terms of the players' mutual beliefs about the basic, objective issues of possible uncertainty, which is equivalent to the existence of a common prior. This condition specifies a countable sequence of questions addressed to the players, which detects when there is no common prior among them.

THE KNOT GROUP AND THE FUNDAMENTAL GROUP OF THE EMBEDDING 3-MANIFOLD

We propose a conjecture concerning 3-manifolds (which implies the ℚ conjecture of Myers') and construct a countable sequence of examples which supports it.

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire spaceNℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ.

Theorems on sets not belonging to algebras

Let A 1 , … , A n , A n + 1 \mathcal {A}_1,\dots , \mathcal {A}_n, \mathcal {A}_{n+1} be a finite sequence of algebras of sets given on a set X X , ⋃ k = 1 n A k ≠ P ( X ) \bigcup _{k=1}^n \mathcal {A}_k \ne \mathfrak {P}(X) , with more than 4 3 n \frac {4}{3}n pairwise disjoint sets not belonging to A n + 1 \mathcal {A}_{n+1} . It has been shown in the author’s previous articles that in this case ⋃ k = 1 n + 1 A k ≠ P ( X ) \bigcup _{k=1}^{n+1} \mathcal {A}_k \ne \mathfrak {P}(X) . Let us consider, instead of A n + 1 \mathcal {A}_{n+1} , a finite sequence of algebras A n + 1 , … , A n + l \mathcal {A}_{n+1}, \dots , \mathcal {A}_{n+l} . It turns out that if for each natural i ≤ l i \le l there exist no less than 4 3 ( n + l ) − l 24 \frac {4}{3}(n+l)- \frac {l}{24} pairwise disjoint sets not belonging to A n + i \mathcal {A}_{n+i} , then ⋃ k = 1 n + l A k ≠ P ( X ) \bigcup _{k=1}^{n+l} \mathcal {A}_k \ne \mathfrak {P}(X) . Besides this result, the article contains: an essentially important theorem on a countable sequence of almost σ \sigma -algebras (the concept of almost σ \sigma -algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function v ( n ) \mathfrak {v}(n) that was introduced by the author in 2002, and other new results.

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds

The main goal of the article is to show that Paley-Wiener functions f ~ L2(M ) of a fixed band width w on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers { ~ i (f)/, i ~ N where { @ i } is a countable sequence of weighted integrals over a collection of "small" and "densely" distributed compact subsets. In particular, {@i }, i N, can be a sequence of weighted Dirac measures ~xi , xi ~ M.

Minima of initial segments of infinite sequences of reals

AbstractSuppose that 〈xk〉k∈ℕ is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA0 suffices to prove the existence of a sequence of reals 〈uk〉k∈ℕ such that for each k, uk is the minimum of {x0, x1, …, xk}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈xk〉k∈ℕ, the stronger subsystem WKL0 is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Limiting partial combinatory algebras

From every partial combinatory algebra A we construct another partial combinatory algebra a-lim( A) . In a-lim( A) , every representable partial numerical function ϕ( n) is exactly of the form lim t ξ( t, n) for some representable partial numerical function ξ( t, n) of A . The partial combinatory algebra a-lim( A) is a quotient of A by a partial equivalence relation, and is equipped with a limit structure in the sense that each element of a-lim( A) is the limit of a countable sequence of A -elements. In this paper, we discuss the limit structures for A in terms of Barendregt's range property (if the range of a combinator is finite, then it is a singleton). Moreover, we repeat the construction a-lim(−) transfinite times to interpret infinitary λ-calculi. Finally, we attempt to interpret type-free λμ-calculus by introducing another partial applicative structure which has an asynchronous application operator that allows a parallel limit operation.

On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac s delta function

The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.

Forcing extensions of partial lattices

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ:ConcK→D be a {∨,0}-homomorphism, where ConcK denotes the {∨,0}-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f:K→L, and an isomorphism α:ConcL→D such that α∘Concf=ϕ. Furthermore, L and f satisfy many additional properties, for example: (i) L is relatively complemented; (ii) L has definable principal congruences; (iii) If the range of ϕ is cofinal in D, then the convex sublattice of L generated by f[K] equals L. We mention the following corollaries, that extend many results obtained in the last decades in that area: 1. Every lattice K such that ConcK is a lattice admits a congruence-preserving extension into a relatively complemented lattice. 2. Every {∨,0}-direct limit of a countable sequence of distributive lattices with zero is isomorphic to the semilattice of compact congruences of a relatively complemented lattice with zero.

Countable infinite sequence of attractors' families for the simplest known equivariant chaotic flow

More than 30 new families of periodic and strange attractors for the simplest known equivariant chaotic jerk equation are studied. Inside each family, both periodic and chaotic attractors result from the subharmonic cascade of a limit cycle born by saddle-node bifurcation. Our numerical results provide clear evidence for the following conjecture: the 35 observed families are the first members of a countable infinite sequence. Furthermore, our simulations point out four power laws relating to the initial infinite sequence of saddle-node bifurcations.

Lattice Fully Orderable Groups

Let Ω be a linearly ordered set, A(Ω) be the group of all order automorphisms of Ω, and L(Ω) be a normal subgroup of A(Ω) consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set Ω such that: (1) A(Ω) is an o-2-transitive group, and (2) Ω contains a countable unbounded sequence of elements, the simple group A(Ω)/L(Ω) has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of A(Ω)/L(Ω) extends to a lattice one (Thm. 2.1). It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group (Thm. 2.2). We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable (Thms. 3.1 and 3.2).

Codimension-onepersistence beyond all orders of homoclinic orbits to singularsaddle centres in reversible systems

The persistence from a singular limit is studied ofhomoclinic orbits to a saddle-centre equilibrium infour-dimensional reversible vector fields.The linearization is assumed tohave eigenvalues ±iω, ±λ, λ,ω>0,with the singular limit being λ→0.Recent work by Lombardi has shown the generic non-persistence of such homoclinic solutions under perturbations thatbreak the integrability of a normal form due to a splitting term(Melnikov function) that is exponentially small in λ/ω.A plausible argument is presented which extends Lombardi's analysisto suggest persistence in a codimension-one sense. Specifically,keeping nonlinear terms fixed but treating λ and ω asindependent parameters, then, under the correct sign of anormal-form coefficient b2, each term in the calculablepart of the Melnikov function has a countable sequence of signchanges as a function ofω. This calculable part of the Melnikov function is only theleading-order term under an additional assumption of closenessto integrability. In that case, a zero of this functionimplies anω-value at which a curve of homoclinic orbits bifurcates fromλ = 0 in the parameter plane. The large-ω asymptotics ofthis sequence of bifurcation points is determined solely by the valueof b2.This heuristic result is supported by careful numerical experiments ona class of fourth-order reversible equations with arbitrary quadraticnonlinear terms. The implications of the results are discussed for solitarywater waves of elevation in the presence of surface tension and forrecently discovered embedded solitons in a variety of(non-integrable) nonlinear optical models.

A σ-ALGEBRA AND A CONCEPT OF LIMIT FOR BODIES

Recent developments in mechanics of continua (the search for optimal shapes of bodies, homogenization theory, the study of the trabecular structure of bones, the dynamics of immiscible mixtures, etc.) render some of the introductory axioms of continuum mechanics inadequate. Not only does one need to give meaning to the join and meet of two bodies, but also to extend the consequent algebra so as to encompass the result of a countable sequence of operations of join or meet; and one should also be able to define the limit of a sequence of bodies. To achieve this goal we propose here to define a body ab initio through the assignment of a probability measure dπ. We realize that π leaves, generally, too much of the texture of the body unspecified; to make up for this deficiency, we suggest the use of appropriate texture measures, reminescent of Tartar's H-measures.9

Ulm Classification of Analytic Equivalence Relations in Generic Universes

AbstractWe prove that if every real belongs to a set generic extension of L, then every Σ equivalence relation E on reals either admits a Δ1 reduction to the equality on the set 2&lt; ω1 of all countable binary sequences, or the Vitali equivalence E0 continuously embeds in E. The proofs are based on a topology generated by OD sets.

An Ulm-type classification theorem for equivalence relations in Solovay model

AbstractWe prove that in the Solovay model, every OD equivalence relation, Ε, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length &lt; ω1) binary sequences, or continuously embeds Ε0, the Vitali equivalence.If Ε is a (resp. ) relation then the reduction above can be chosen in the class of all Δ1 (resp. Δ2) functions.The proofs are based on a topology generated by OD sets.

On uncountable cardinal sequences for superatomic Boolean algebras

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results: (1) Ifθ = 〈κα :α <ω1〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB. (2) Ifκ is an uncountable cardinal such thatκ<κ =κ andθ = 〈κα :α <κ+〉 is a cardinal sequence such thatκα ≥κ for everyα <κ+ andκα =κ for everyα <κ+ such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.