Perfect Subset Research Articles

Monotone operators and first category sets

In this paper we show that the local monotonicity in the sense of Minty and Browder on some residual sets assure the global monotonicity and, according to an earlier result, the convexity of the inverse images. We pay some special attention to the residual sets arising as complements of some special first Baire category sets, namely the \(\sigma \)-affine sets, the \(\sigma \)-compact sets and the \(\sigma \)-algebraic varieties. We achieve this goal gradually by showing, at first, that the continuous real valued functions of one real variable, which are locally nondecreasing on sets whose complements have no nonempty perfect subsets, are globally nondecreasing. The convexity of the inverse images combined with their discreteness, in the case of local injective operators, ensure the global injectivity. Note that the global monotonicity and the local injectivity of regular enough operators is guaranteed by the positive definiteness of the symmetric part of their Gâteaux differentials on the involved residual sets. We close this work with a short subsection on the global convexity which is obtained out of its local counterpart on some residual sets.

Distinguishing classical correlations from quantum correlations

The aim of this paper is to distinguish classical correlations from quantum ones. First, a new characterization of a classical correlated (CC) state is established. Corresponding results for the left and right classical correlations are also obtained. Second, a sufficient and necessary condition for a convex combination of two CC states to be CC is given. It is proved that the set CC of all CC states on is a perfect, nowhere dense and compact subset of the metric space of all states (density operators) on . Based on our new characterization of CC states, a quantity Q(ρ) is associated with a state ρ. It is shown that a state ρ is CC if and only if Q(ρ) = 0. In particular, we prove that the Werner state Wλ in a two-qubit system with a single real parameter λ is CC if and only if λ = 0.25.

On three questions concerning groups with perfect order subsets

On three questions concerning groups with perfect order subsets

A dedekind finite borel set

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if \({B\subseteq 2^\omega}\) is a G δσ -set then either B is countable or B contains a perfect subset. Second, we prove that if 2ω is the countable union of countable sets, then there exists an F σδ set \({C\subseteq 2^\omega}\) such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite \({D\subseteq 2^\omega}\) which is F σδ .

Performance of super-orthogonal space–time trellis codes with transmit antenna selection

Super-orthogonal space-time trellis (SOSTT) codes provide a significant coding gain with respect to the conventional space-time trellis (STT) codes without increasing the decoding complexity at the receiver. Transmit antenna selection (TAS) is an important technique to solve the implementation complexity problems of multiple-antenna systems that arise from employing a separate RF chain for each antenna. This study treats a scheme combining SOSTT coding with TAS in order to exploit performance improvements provided by SOSTT codes while realising the promises made by the TAS technique. In the proposed scheme, transmit antennas that maximise the signal-to-noise ratio at the receiver are chosen and activated out of all available transmit antennas for transmission of the baseline SOSTT code, whereas all other transmit antennas are silent. The error performance of the proposed structure is investigated by deriving moment generating function (MGF)-based upper bound expressions on the pairwise error probability in quasi-static flat Rayleigh fading channels for both perfect and imperfect antenna subset selection. The performance analysis demonstrates that the proposed scheme provides full diversity order as if all transmit antennas were used when antennas are perfectly selected. Moreover, it is shown that this structure provides better error performance almost at the same decoding complexity compared to the previous STT coding structures with TAS in the literature.

An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform a ↦ a ˆ | E from L 1 ( R ) into C ( E ) is onto. Helson proved that then, for μ ∈ M ( E ) , lim | y | → ∞ | μ ˆ ( y ) | = 0 is possible only if μ = 0 . In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.

First Countable Continua and Proper Forcing

Abstract.Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight ℵ1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Low-distortion embeddings of infinite metric spaces into the real line

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K > 1 every uncountable Polish space has a perfect subset that K -bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K > 1 , K -bi-Lipschitz embed into the real line.

Thin Sets with Fat Shadows: Projections of Cantor Sets

A Cantor set is a nonempty, compact, totally disconnected, perfect subset of IR. Now, the set being totally disconnected means that it is scattered about like a “dust”. If you shine light on a clump of dust floating in the air, the shadow of this dust will look like a bunch of spots on the wall. You would be very surprised if you saw that the shadow was a filled-in shape (like a rabbit, say!). That would be pretty unbelievable. So, is this possible? We can think of the projection of a Cantor set onto a subspace as the shadow on that subspace. Is it possible that a cloud of dust (a Cantor set) could have a shadow (projection) which is “filled-in” (homeomorphic to the n − 1 dimensional unit ball)? The answer is YES! In fact, it is possible to have the shadow in every direction be “filled-in”! In this note we give an example of a simple construction of a Cantor subset of the unit square whose projection in every direction is a line segment. This construction can easily be generalized to n dimensions.

Hurewicz sets of reals without perfect subsets

We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice function whose image is a gamma-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszynski and Tsaban, and implies that for C_p(X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangelskii's property alpha_2.

Network properties of an epiphyte metacommunity

Summary 1 Trophic relationships are often depicted as networks, in which species are connected by links representing predatory interactions. Network analyses have also been applied to non-trophic species interactions, such as pollination and seed dispersal mutualisms, to summarize community-level patterns in mutually beneficial interactions. However, the processes responsible for common network properties such as degree distributions (the number of interactions maintained by each species in the network) and nestedness (the tendency for specialists to interact with perfect subsets of the species interacting with generalists) are poorly understood. 2 I evaluated the network properties of a novel type of species interaction, commensal interactions between epiphytes and their host trees. I quantified the occurrence of 19 vascular epiphyte species on seven host tree species in a metacommunity of epiphytes in New Zealand. I then developed null models to test whether observed network properties result from the frequency of species interactions, which is a commonly cited but rarely tested explanation for degree distributions and nestedness. 3 Results showed that degree distributions of both epiphyte and host tree species were consistent with a null model that randomly populated hosts with epiphytes. Several null models that test for nestedness by randomly linking pairs of interacting species showed significantly higher nestedness in the real community. In fact, the observed level of nestedness was among the highest yet recorded for any type of ecological interaction. A second null model that randomized species interactions according to their overall frequencies of occurrence in the community confirmed support for nestedness. 4 Results indicate that the degree distributions were consistent with randomized expectations; the number of interactions maintained by both epiphyte and host tree species appears to be determined by their frequency of occurrence in the community. However, interaction frequencies could be ruled-out as an explanation for nestedness. Nested epiphyte–host interactions may instead result from a predictable sequence of epiphyte succession, wherein earlier colonists ameliorate environmental conditions within host trees for later recruiting species.

Singular inner functions whose Frostman shifts are Carleson–Newman Blaschke products II

Let be the family of inner functions whose non-trivial Frostman shifts are Carleson–Newman Blaschke products. It is known that for any closed subset of the unit circle ∂D, there is a discrete singular inner function S μ with supp (μ)=E and . In this article, we are interested in continuous singular inner functions S μ in or not in with nonporous supp (μ). For example, if E is a perfect subset of ∂D, then there is a continuous singular inner function with supp (μ)=E (Theorem 2.6). We also show that if E is a perfect subset of ∂D which is not porous, then there is a continuous singular inner function with supp (μ)=E (Corollary 3.5).

A class of Banach algebras whose duals have the Radon-Nikodym property

Let A be a complex, commutative Banach algebra and let M A be the structure space of A. Assume that there exists a continuous homomorphism h : L1(G) → A with dense range, where L1(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M A is scattered (i.e., it has no nonempty perfect subset) and $$A^{*} \cdot A = \overline{{{\text{span}}}} M_{A}$$ . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if $$A^{*} = \overline{{{\text{span}}}} M_{A}$$ . Furthermore, any of these conditions implies that M A is scattered. Several applications are given.

Parametrizing the abstract Ellentuck theorem

We give a parametrization with perfect subsets of 2 ∞ of the abstract Ellentuck theorem (see [T.J. Carlson, S.G. Simpson, Topological Ramsey Theory, in: J. Neŝetrˆil, V. Rödl (Eds.), Mathematics of Ramsey Theory, Algorithms and Combinatorics, vol. 5, Springer, Berlin, 1990, pp. 172–183], [S. Todorcevic, Introduction to Ramsey spaces, to appear] or [S. Todorcevic, Lecture notes from a course given at the Fields Institute in Toronto, Canada, Autumn 2002]). The main tool for achieving this goal is a sort of parametrization of an abstract version of the Nash–Williams theorem. As corollaries, we obtain some known classical results like the parametrized version of the Galvin–Prikry theorem due to Miller and Todorcevic [A.W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic 41 (1989) 179–203], and the parametrized version of Ellentuck's theorem due to Pawlikowski [Parametrized Ellentuck theorem, Topology Appl. 37 (1990) 65–73]. Also, we obtain parametized vesions of nonclassical results such as Milliken's theorem [K.R. Milliken, Ramsey's theorem with sums or unions, J. Combin. Theory (A) 18 (1975) 276–290], and we prove that the family of perfectly Ramsey subsets of 2 ∞ × FIN k [ ∞ ] is closed under the Souslin operation.

Semisimplicity of Some Class of Operator Algebras on Banach Space

Let G be a locally compact abelian group and let \({\bf T}{\text{ = }}{\left\{ {T{\left( g \right)}} \right\}}_{{g \in G}} \) be a representation of G by means of isometries on a Banach space. We define WT as the closure with respect to the weak operator topology of the set \({\left\{ {\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)}:f \in L^{1} {\left( G \right)}} \right\}}, \) where \(\ifmmode\expandafter\hat\else\expandafter\^\fi{f}{\left( {\text{T}} \right)} = {\int\limits_G {f{\left( g \right)}T{\left( g \right)}dg} } \) is the Fourier transform of f ∈L1(G) with respect to the group T. Then WT is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra WT is semisimple.

Mass Problems and Randomness

AbstractA mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices ω and s of weak and strong degrees of mass problems given by nonempty subsets of 2ω. Using an abstract Gödel/Rosser incompleteness property, we characterize the subsets of 2ω whose associated mass problems are of top degree in ω and s, respectively Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within ω. Namely, we characterize r as the unique largest weak degree of a subset of 2ω of positive measure. Within ω we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect subsets of 2ω. In addition, we present other natural examples of intermediate degrees in ω. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

On decompositions of Banach spaces of continuous functions on Mrówka’s spaces

It is well known that if K K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space C ( K ) C(K) of continuous functions on K K has complemented copies of c 0 c_{0} , i.e., C ( K ) ∼ c 0 ⊕ X ∼ c 0 ⊕ c 0 ⊕ X ∼ c 0 ⊕ C ( K ) C(K)\sim c_{0} \oplus X\sim c_{0}\oplus c_{0} \oplus X\sim c_{0}\oplus C(K) . We address the question if this could be the only type of decompositions of C ( K ) ≁ c 0 C(K)\not \sim c_{0} into infinite-dimensional summands for K K infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin’s axiom we construct an example of Mrówka’s space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

Separately Nowhere Constant Functions; n-Cube and α-Prism Densities

A function ƒ from a countable product X = ΠiXi of Polish spaces into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of Xi's. This result is used to distinguish between n-cube density notions for different n ≤ ω, where ω-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of α < ω1, between the notions of α-prism densities — the more refined density notions used also in CPA.

Uniformly Perfect Subsets of the Real Line and John Domains

Let E be a regular compact subset of the real line. We relate the Green function for the complement of E (with respect to the extended complex plane) to some conformal map f. We establish the relationship between the geometry of E and f(E).

Extreme points in spaces of polynomials

We determine the extreme points of the unit ball in spaces of complex polynomials (of a fixed degree), living either on the unit circle or on a subset of the real line and endowed with the supremum norm. Introduction Let Pn stand for the space of all polynomials with complex coefficients of degree not exceeding n. Given a compact set E ⊂ C, one may treat Pn as a subspace of C(E), the space of continuous functions on E, and equip it with the maximum norm ‖P‖∞ = ‖P‖∞,E := max z∈E |P (z)| (P ∈ Pn). The resulting space will be denoted by Pn(E). We write ball(Pn(E)) := {P ∈ Pn : ‖P‖∞,E ≤ 1} for the unit ball of Pn(E), and we shall be concerned with the extreme points of this ball. (As usual, an element of a convex set S is said to be its extreme point if it is not the midpoint of any nondegenerate segment contained in S.) In this paper, we explicitly characterize the extreme points of ball(Pn(E)) in the case where E is either the circle T := {z ∈ C : |z| = 1} or a perfect compact subset of the real line R. The description obtained is, perhaps, a bit more complicated than one could at first expect; however, the complexity seems to be in the nature of things. Let us begin by recalling that the extreme points of the unit ball in L∞(T) – or in C(T) – are precisely the functions of modulus 1. (The same applies to other sets in place of T.) Further, in the space H∞ of bounded analytic functions on {|z| < 1}, as well as in the disk algebra H∞ ∩ C(T), the extreme points are known to be the unit-norm functions f with ∫ T log(1 − |f(z)|2)|dz| = −∞; see [H, Chap. 9]. Received September 30, 2002. 2000 Mathematics Subject Classification. 46B20, 46E30 Supported in part by Grant 02-01-00267 from the Russian Foundation for Fundamental Research, by a PIV fellowship from Generalitat de Catalunya, and by the Ramon y Cajal program (Spain).