Solution of a Problem of L. Fuchs Concerning Finite Intersections of Pure Subgroups

1986 ◽  
Vol 38 (2) ◽  
pp. 304-327 ◽  
Author(s):  
R. Göbel ◽  
R. Vergohsen
Keyword(s):  
Finite Number ◽  
Abelian Groups ◽  
Natural Numbers ◽  
P 75 ◽  
Image Position ◽  
Special Case

L. Fuchs states in his book “Infinite Abelian Groups” [6, Vol. I, p. 134] the followingProblem 13. Find conditions on a subgroup of A to be the intersection of a finite number of pure (p-pure) subgroups of A.The answer to this problem will be given as a special case of our theorem below. In order to find a better setting of this problem recall that a subgroup S ⊆ E is p-pure if pnE ∩ S = pnS for all natural numbers. Then S is pure in E if S is p-pure for all primes p. This generalizes to pσ-isotype, a definition due to L. J. Kulikov, cf. [6, Vol. II, p. 75] and [11, pp. 61, 62]. If α is an ordinal, then S is pσ-isotype if

FC-nilpotent and FC-soluble groups

1956 ◽  
Vol 52 (3) ◽  
pp. 391-398 ◽  
Author(s):  
A. M. Duguid ◽  
D. H. McLain
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Abelian Groups ◽  
Characteristic Subgroup ◽  
Arbitrary Group ◽  
Soluble Groups ◽  
Image Position

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

Frequency computations and the cardinality theorem

1992 ◽  
Vol 57 (2) ◽  
pp. 682-687 ◽  
Author(s):  
Valentina Harizanov ◽  
Martin Kummer ◽  
Jim Owings
Keyword(s):  
Characteristic Function ◽  
Recursive Function ◽  
Natural Numbers ◽  
New Techniques ◽  
Semirecursive Set ◽  
Image Position ◽  
Special Case ◽  
Full Proof

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.

scholarly journals On the metric theory of the optimal continued fraction expansion

1997 ◽  
Vol 56 (1) ◽  
pp. 69-79
Author(s):  
R. Nair
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Natural Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Special Case

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants byIn this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

scholarly journals THE DP-RANK OF ABELIAN GROUPS

2019 ◽  
Vol 84 (3) ◽  
pp. 957-986 ◽  
Author(s):  
YATIR HALEVI ◽  
DANIEL PALACÍN
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Natural Numbers ◽  
Image Position ◽  
Stable Field

AbstractAn equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p, there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

On the estimation of a harmonic component in a time series with stationary dependent residuals

1973 ◽  
Vol 5 (02) ◽  
pp. 217-241 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Harmonic Component ◽  
Asymptotic Distributions ◽  
Finite Variance ◽  
Rigorous Derivation ◽  
Image Position ◽  
Vector Valued ◽  
Special Case

Let observations (X 1, X 2, …, Xn ) be obtained from a time series {Xt } such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu (θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu (θ) = 0 for u > 0, the parameter θ thus being absent.

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

Proof that there are infinitely many modalities in Lewis's system S2

1940 ◽  
Vol 5 (3) ◽  
pp. 110-112 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Finite Number ◽  
Infinite Matrix ◽  
Image Position

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then

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