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.

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

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.

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

scholarly journals Survival Probabilities Based on Pareto Claim Distributions: Comment

1980 ◽  
Vol 11 (2) ◽  
pp. 154-157 ◽  
Author(s):  
Marc J. Goovaerts ◽  
Nelson de Pril
Keyword(s):  
Characteristic Function ◽  
Density Function ◽  
Pareto Distribution ◽  
Beta Function ◽  
The Other ◽  
Survival Probabilities ◽  
Image Position ◽  
Derivatives Of ◽  
Special Case

In a recent paper Seal (1980) calculated numerically survival probabilities based on Pareto claim distributions.The Pareto density may be written asGeneralizing, the Pareto distribution may be regarded as a special case of the so-called beta-prime distribution (Keeping, 1962, p. 83) with density functionwhere B(p, q) = is the beta function.In his paper Seal (1980, Appendix 1) arrived at a contradiction concerning this beta-prime distribution. He found on one side that all derivatives of the characteristic function exist at the origin and on the other side that only the moments of order n < q exist. In this note we will show that this contradiction is due to the use of an incorrect expression for the characteristic function of the beta-prime distribution, which was taken over from Johnson and Kotz (1970, Ch. 26) and Oberhettinger (1973, Table A).

The complexity of ODDnA

2000 ◽  
Vol 65 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Richard Beigel ◽  
William Gasarch ◽  
Martin Kummer ◽  
Georgia Martin ◽  
Timothy Mcnicholl ◽  
...  
Keyword(s):  
Characteristic Function ◽  
Lower Bounds ◽  
Truth Table ◽  
Query Complexity ◽  
Recursively Enumerable ◽  
Fixed Set ◽  
Recursively Enumerable Sets ◽  
Semirecursive Set ◽  
Parallel Queries ◽  
Image Position

AbstractFor a fixed set A. the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A:and, for m > 2,where #nA. (x1….. xn) = A(x1) + A(xn)(We identify with , where χA is the characteristic function of A.)If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODDnA and MODmnA:• ODDnA can be decided with n parallel queries to A, but not with n − 1.• ODDnA can be decided with ⌈log(n + 1)⌉ sequential queries to A but not with ⌈log(n + 1)⌉ − 1.• MODmnA can be decided with ⌈n/m⌉ + ⌊n/m⌋ parallel queries to A but not with ⌈n/m⌉ + ⌊n/m⌋ − 1.• MODmnA can be decided with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ − 1.The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.)In particular, every nonzero truth-table degree contains a set A such that ODDnA cannot be decided with n − 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDnB can be decided with one query to B, a set's query complexity depends more on its structure than on its degree.For a fixed set A,Q(n, A) = {S: S can be decided with n sequential queries to A}.Q∥ (n, A) = {S : S can be decided with n parallel queries to A}.We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies:• Q(0,A) ⊂ Q (1, A) ⊂ Q(2, A) ⊂ …Q∥ (0, A) ⊂ Q∥ (1, A) ⊂ Q∥ (2, A) ⊂ …The same is true if A is a jump.

A note on the representation of general recursive functions and the μ quantifier

1959 ◽  
Vol 55 (2) ◽  
pp. 145-148
Author(s):  
Alan Rose
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Image Position ◽  
Primitive Recursive

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schemasubject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

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 &gt; 0, the parameter θ thus being absent.

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