Cardinality Of The Continuum Research Articles

Applications of linear functionals to Kronecker modules. II

This paper continues the investigation of a class of modules E r ⩾ 2, constructed from submodules of the module of rational functions K( X), over the polynomial ring K[ X], given by a height function h; sequences of linear functionals L=( l 2,…, l r on K( X); and a sequence of positive integers ( m 2,…, m r . The emphasis is on when 1<card{θϵ K ∪ {∞}: h(θ)≠}<∞. To each subset L′ of { l 2,…, l r } is attached a field extension K L′ of the prime subfield of K. Let M L 1 and M L 2 be modules constructed from the sequences of linear functionals L 1 and L 2. If M L 1 is isomorphic to M L 2, then K L 1 K L 2 are contained in finitely generated field extensions of each other. Let L̄ i = L ⧹ l i , i=2,…, r. If M L is not purely simple, then for some i in { 2,…, r}, K l i is contained in a finitely generated extension of K L̄ i . These results are then used to obtain the following theorems: assuming that 1<supp h<∞, then there are exactly ℵ 0 isomorphism classes of completely decomposable modules in E r ; there are at least c isomorphism classes of (i) indecomposable but not purely simple modules in E r , (ii) purely simple modules in E r ( c = the cardinality of the continuum).

POWER SERIES AND PEANO CURVES

A series with coefficients tending monotonically (decreasing) to zero is constructed whose sum has the following property: for any complex number the set has the cardinality of the continuum. Here the domain contains the segment on both the real and the imaginary axes. The construction is based on corresponding properties of lacunary trigonometric series, which are presented in detail. Bibliography: 8 titles.

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx eZ, letf |n| denote then'th iterate off. Clearlyf |m|(f |n|(x))=f |m+n|(x) for allm,neZ and allxeR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxeR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F |α|:αeG} on the real line with the properties (i)F |α|(F |β|(x))=F |α+β| (x) for allxeR and (ii)F |n|(x)=f |n|(x) for allneZ andxeR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.

Existentially closed structures in the power of the continuum

On introduit la notion de systeme arbre. Sous de bonnes conditions on peut utiliser les systemes arbres pour construire des structures de la puissance du continu qui sont ∞ ω -equivalentes a une structure denombrable prescripte

Toeplitz minimal flows which are not uniquely ergodic

We study minimal symbolic dynamical systems which are orbit closures of Toeplitz sequences. We construct 0–1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality, is countably infinite or has cardinality of the continuum.

On Partitions of Plane Sets Into Simple Closed Curves. II

We answer some questions raised in [1]. In particular, we prove: (i) Let $F$ be a compact subset of the euclidean plane ${E^2}$ such that no component of $F$ separates ${E^2}$. Then ${E^2}\backslash F$ can be partitioned into simple closed curves iff $F$ is nonempty and connected. (ii) Let $F \subseteq {E^2}$ be any subset which is not dense in ${E^2}$, and let $\mathcal {S}$ be a partition of ${E^2}\backslash F$ into simple closed curves. Then $\mathcal {S}$ has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.

Large families of incomparable A-isols

The set Λ of isols was extensively studied by Dekker and Myhill in [1]. Subsequently, Nerode [3] developed the theory of Λ(A), the set of isols relative to some recursively closed set of functions A.One of the main areas of interest of [1] was the natural partial order ≤ on Λ. In this paper we will examine some of the properties of ≤A on Λ(A). We use the following notations: ∣A∣ is the cardinality of the set A, ⊃ denotes strict inclusion, (a) is the power set of the set a, c is the cardinality of the continuum, and ω = {0, 1, 2, …}. The terms A-isol, A-immune, A-r.e., A-incomparable, etc. all refer to the usual meaning of these words, only taken in the context of the recursively closed set A. ReqA(a) is the A-r.e.t. of which a is a representative. By identifying a finite natural number with the A-r.e.t. consisting of sets of a given finite cardinality we see that ω ⊆ Λ(A); Λ(A) is said to be nontrivial iff ω ⊃ Λ(A). The three results proven in this paper are:Theorem 1. If Λ(A) is nontrivial, then ∣Λ(A)∣ = c.Theorem 2. If∣A∣ &lt; c, then Λ(A) is nontrivial.Theorem 3. If ∣A∣ &lt; c and ∣⊿∣ &lt; c and ⊿ ⊆ Λ(A) − ω, then there is aΓ ⊆ Λ(A) − ω such that:(a) ∣Γ∣ = c.(b) Every member of Γ is A-incomparable with every member of Δ.(c) Any two distinct members of Γ are A-incomparable.

Recursive equivalence types and octahedra

AbstractLet the word “graph” be used in the sense of a countable, connected, simple graph with at least one vertex. We write Qn and Ocn for the graphs associated with the n-cube Qn and the n-octahedron Ocn respectively. In a previous paper (Dekker, 1981) we generalized Qn and Qn to a graph QN and a cube QN, for any nonzero recursive equivalence type N. In the present paper we do the same for Ocn and Ocn. We also examine the nature of the duality between QN and OcN, in case N is an infinite isol. There are c RETs, c denoting the cardinality of the continuum.

On partitions of plane sets into simple closed curves. II

We answer some questions raised in [1]. In particular, we prove: (i) Let F F be a compact subset of the euclidean plane E 2 {E^2} such that no component of F F separates E 2 {E^2} . Then E 2 ∖ F {E^2}\backslash F can be partitioned into simple closed curves iff F F is nonempty and connected. (ii) Let F ⊆ E 2 F \subseteq {E^2} be any subset which is not dense in E 2 {E^2} , and let S \mathcal {S} be a partition of E 2 ∖ F {E^2}\backslash F into simple closed curves. Then S \mathcal {S} has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.

Twilight graphs

AbstractThis paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite:(a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent,(b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them.A graph G = ‹ν, η› with ν as set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs G = ‹ν, η› with n = card(ν), e = card(η), do generalize to isolic ω-graphs, e.g., n − 1 ≤ e ≤ n(n − 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree G = ‹ν, η›: N = E + 1 iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs.The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph.An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum.

No system of uncountable rank is purely simple

A pair of complex vector spaces (V, W) is a system if and only if there is a C-bilinear map C 2 × V {{\mathbf {C}}^2} \times V to W. The category of systems is equivalent to the category of modules over a certain subring of the ring of 3 × 3 3 \times 3 matrices over the complex numbers, and so module-theoretic concepts make sense for systems. A system is purely simple if it has no proper pure subsystem. Recently it has been shown that for every positive integer n, there exists a purely simple system of rank n but no system of rank greater than the cardinality of the continuum is purely simple. In this paper it is shown that no system of rank greater than ℵ 0 {\aleph _0} is purely simple. Necessary and sufficient conditions for a system of rank ℵ 0 {\aleph _0} to be purely simple are also given.

Set of asymptotic values of a meromorphic function of finite order

It is well known that an entire function of finite order has an at most countable number of asymptotic values. Gross [I] constructed an example of an entire function of infinite order whose set of asymptotic values coincides with the extended complex plane. Dreisin and Weizmann [2] raised the question of whether there are any restrictions from above on the size of the set of asymptotic values of a meromorphic function of finite order. In this direction only the result of Valiron [3] is known; he constructed a meromorphic functlon of finite order with a set of asymptotic values having the cardinality of the continuum 9 In this paper we construct a meromorphic function of finite order whose set of asymptotic values coincides with the extended complex plane, thereby giving a negative answer to the question of Dreisin and Weizmann.

A compact space having the cardinality of the continuum with no convergent sequences

1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.(2) Every compact space of cardinality ≤ ʗ is sequentially compact.(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.

Semivarieties and Schreier varieties of unary algebras

A correction is provided for one result of a paper by Shafaat [1], and it is proven that the set of Schreier varieties of algebras 〈A;+1〉 with two unary operations has the cardinality of the continuum.

ON THE PROBLEM OF A FINITE BASIS OF IDENTITIES IN GROUPS

This paper contains a proof that the set of varieties of groups has the cardinality of the continuum. This implies the existence of an infinite system of group identities not equivalent to any finite system.

Countable vector spaces with recursive operations Part I1

We use the word “number” for “nonnegative integer” and “set” for collection of numbers. The set of all numbers is denoted by ε and the empty set by ο. We write ⊂ for the relation of inclusion and ⊆ for that of proper inclusion. If ƒ is a function from a subset of ε into ε, its domain and range are denoted by δƒ and ρƒ respectively. For any collection Γ of entities, card Γ or card (Γ) stands for the cardinality of Γ. The cardinality of the continuum is denoted by c.

𝑚-bounded uniformities between two given uniformities

Introduction. This paper is an attempt to gain some information about the lattice of the uniformities which can be defined on a set X. Our results are stated in terms of the m-boundedness of uniformities, where m is an infinite cardinal. We define a pseudo-metric p on X to be m-bounded if for e > 0, X can be written as a union of fewer than m sets, each of p-diameter not exceeding E. If P is a family of m-bounded pseudo-metrics on X we say that the associated uniformity &(P) is m-bounded. We call &(P) strictly m-bounded if m is its least (infinite) bound. Roughly speaking, we found that if two uniformities 9 and 91 have different strict bounds, with 90c l, then there are at least c uniformities between them. (Here c is the cardinality of the continuum.) The lower bound on the number of uniformities in-between can be improved as the strict bound on 91 increases. Moreover, these uniformities can be chosen to be strictly m-bounded, provided m exceeds the bound on 90 (but not the bound on Sb). These results throw some light on the structure of a proximity class. For example, if a proximity class has a non-totally-bounded member then it must contain at least c uniformities. If a proximity class has a member which is strictly m-bounded then for every cardinal n between No and m the class has a strictly n-bounded member (in fact, quite a few). Thus, no gaps are allowed as far as n-boundedness is concerned. Finally, no proximity class has a minimal m-bounded uniformity, for mr>NO. We would like to express our thanks to the referee for simplifying our proofs and our results, and indeed for strengthening our results. From his improvements we were able to obtain further extensions. Theorem 2.1 is in fact a combined effort.

A note on a paper of Paul Hill and Charles Megibben

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with [p] the same subgroup of G is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

Density of one graph along another

Introduction. The word shall mean the graph of a real function, and the X-projection of a graph F is the set of all abscissas of points of F. c denotes the cardinality of the continuum. If H is a collection of sets, H* denotes the union of all the sets in H. DEFINITION. Suppose F and G are graphs with X-projection [0, 1]. The statement that F is dense (c-dense) along G means that if [a, b] is a subinterval of [0, 1], then there is a point (are c-many points) of intersection of F and G with abscissa in [a, b]. In this paper, the following three theorems will be proved:

Central Limit Theorems for Families of Sequences of Random Variables

Let $F$ be a (nonempty) set of distribution functions (d.f.'s) of random variables (r.v.'s) with zero means and positive, finite variances. Let $\mathfrak{F}(F)$ be the set of all sequences of independent r.v.'s (independent within each sequence) whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. No assumptions are made on the interrelations between the joint probability spaces of the r.v.'s of different sequences of $\mathfrak{F}(F)$. Accordingly, dependence or independence between r.v.'s of different sequences needs not be specified. A generic member of $\mathfrak{F}(F)$ will be denoted by $\epsilon = \{\epsilon_k; k = 1, 2, \cdots\}$, or, when we discuss sequences of members of $\mathfrak{F}(F)$, by $\epsilon(n) = \{\epsilon_{nk}; k = 1, 2, \cdots\}, n = 1, 2, \cdots$. In the following, $\mathfrak{F}(F)$ plays the role of a parameter space, the parameter points being $\epsilon, \epsilon(1), \epsilon(2), \cdots$. Since only the d.f.'s of a sequence of (independent) r.v.'s are relevant to the central limit theorem, instead of $\mathfrak{F}(F)$, the set of sequences of d.f.'s corresponding to the elements of $\mathfrak{F}(F)$ may also be regarded as the parameter space. It clearly is a map of $\mathfrak{F}(F)$. The inverse mapping subdivides $\mathfrak{F}(F)$ into certain equivalence classes. From elementary set theory it follows that $\mathfrak{F}(F)$ as well as its map has the cardinality of the continuum if $F$ contains more than one element. Further, let $\{a_{nk}; n = 1, 2, \cdots; k = 1, 2, \cdots\}$ be a double sequence of real constants, and let $\{k_n\}$ be a sequence of positive integers such that $a_{nk_n} \neq 0$ and $a_{nk} = 0$ for $k > k_n, n = 1, 2, \cdots$. We denote the variance of $\sum^{k_n}_{k=1} a_{nk}\epsilon_{nk}$ by \begin{equation*}\tag{1}B_n^2 = \sum^{k_n}_{k=1} a^2_{nk}\sigma^2_{nk}, \quad\sigma^2_{nk} = \operatorname{var} \epsilon_{nk},\end{equation*} and put \begin{equation*}\tag{2}\zeta_n = B_n^{-1} \sum^{k_n}_{k=1} a_{nk}\epsilon_{nk},\quad n = 1, 2, \cdots,\end{equation*} where $B_n = + (B_n^2)^{\frac{1}{2}}$. If all sequences $\epsilon(n)$ are identical we denote them by $\epsilon$ and write instead of (2) $\zeta_n(\epsilon) = B_n^{-1}(\epsilon) \sum^{k_n}_{k=1} a_{nk}\epsilon_k, \quad n = 1, 2, \cdots,$ which shows the dependence on the parameter $\epsilon$ more clearly; $B_n^2(\epsilon)$ is defined analogously to (1). This note deals with necessary and sufficient conditions on the set $F$ and on the double sequence $\{a_{nk}\}$ in order that the d.f.'s of the $\zeta_n$ tend to the standard normal d.f. (denoted by $\Phi(x)$) and that the $B^{-1}_na_{nk}\epsilon_{nk}$ are infinitesimal for every sequence of sequences $\epsilon(1), \epsilon(2), \cdots \varepsilon \mathfrak{F}(F)$ (Theorem 1). For the case of identical sequences $\epsilon(n) (= \epsilon)$, which is of particular interest in applications, a normal convergence theorem (Theorem 3) holding uniformly for $\epsilon$ on $\mathfrak{F}(F)$ is obtained under the same necessary and sufficient conditions as in Theorem 1. Theorem 3 yields, e.g., conditions for the asymptotic normality of the least squares estimators of the parameters in a linear regression with independent and not necessarily identically distributed errors whose d.f.'s are unknown but belong to a certain class $F$ (Eicker (1963)). From the necessity of the conditions it follows that these conditions are the best possible ones under the limited information about the error terms provided by the model assumptions. Frequently, when limit theorems for families of sequences of random variables are met in statistics and probability theory, the emphasis is on the uniformity of the convergence of the sequences with respect to the family parameter which assumes values in a given a priori set. (For an example, compare Parzen (1954), p. 38. That paper also cites some of the earlier publications on the subject.) The present note emphasizes not primarily the uniformity of the convergence, but the necessity of the conditions (including one on the parameter space $\mathfrak{F}(F))$ for the convergence on the parameter space. Accordingly, the set $F$ cannot be prescribed arbitrarily. The uniformity of the convergence on $\mathfrak{F}(F)$ is shown without difficulty to be implied by the ordinary convergence on $\mathfrak{F}(F)$. Another difference from earlier work may be seen in the particular structure of $\mathfrak{F}(F)$, which is comparable to an infinite-dimensional vector space. Although many of the earlier convergence theorems are (or can be) formulated for abstract parameter spaces, in the applications such as estimation theory these spaces are usually specialized to intervals on the line or in a finite-dimensional vector space.