Meager Subsets Research Articles

Porosities of the sets of attractors

This paper is another attempt to measure the difference between the family A[0,1] of attractors for iterated function systems acting on [0,1] and a broader family, the set Aw[0,1] of attractors for weak iterated function systems acting on [0,1]. It is known that both A[0,1] and Aw[0,1] are meager subsets of the hyperspace K([0,1]) (of all compact subsets of [0,1] equipped with the Hausdorff metric). Moreover, A[0,1] is σ-lower porous while the question about σ-lower porosity of Aw[0,1] is still open. We prove that A[0,1] is not σ-strongly porous in K([0,1]). Moreover, we show that Aw[0,1]∖A[0,1] is dense in K([0,1]).

A Fuzzy based technique for Pattern Recognition & Classification

A pattern can either be seen genuinely or it tends to be watched numerically by applying calculations. Pattern classification is worried about the capacity to discover absolute names for a lot of perceptions. A pattern classification task was viewed as an example determination issue where a meager subset of test from the marked preparing set was picked. We proposed a versatile learning calculation using the least square capacity to address this issue. Utilizing these chose tests, which we call educational vectors, a classifier equipped for perceiving the test tests was built up. This epic calculation is a mix of looking through systems that, in light of forward looking through advances, yet Adaptive finds a way to address the blunders presented by before forward advances. This paper reviews cost-delicate fuzzy standard based frameworks for pattern classification. Weighted preparing patterns are utilized to build cost-touchy fuzzy principle-based frameworks. A fuzzy classification framework is built from a given arrangement of preparing patterns. It is accepted that a weight is appointed to each preparation pattern from the earlier. The heaviness of preparing patterns can be determined dependent on their dispersion.

Superintegrable Bertrand Natural Mechanical Systems

The problem of finding superintegrable systems (i.e., systems with closed trajectories in a certain domain) in the class of natural mechanical systems invariant under rotations goes back to the works of Bertrand and Darboux. Systems of Bertrand type under various restrictions were described by J. Bertrand, G. Darboux, V. Perlik, A. Besse, O. A. Zagryadsky, E. A. Kudryavtseva, and D. A. Fedoseev. However, in full generality, this issue remained open because of the so-called equator problem. In the remaining difficult case with equators, we describe all Bertrand natural mechanical systems and also solve the problem on the connection between various classes of systems of the Bertrand type (the widest class of locally Bertrand systems, the class of Bertrand systems, the narrow class of strongly Bertrand systems, and so on), which coincide in the previously studied case of configuration manifolds without equators. In particular, we show that strongly Bertrand systems form a meager subset in the set of Bertrand systems, and Bertrand systems form a meager subset in the set of locally Bertrand systems.

𝜎-ideals and outer measures on the real line

Abstract A weak selection on ℝ {\mathbb{R}} is a function f : [ ℝ ] 2 → ℝ {f\colon[\mathbb{R}]^{2}\to\mathbb{R}} such that f ⁢ ( { x , y } ) ∈ { x , y } {f(\{x,y\})\in\{x,y\}} for each { x , y } ∈ [ ℝ ] 2 {\{x,y\}\in[\mathbb{R}]^{2}} . In this article, we continue with the study (which was initiated in [1]) of the outer measures λ f {\lambda_{f}} on the real line ℝ {\mathbb{R}} defined by weak selections f. One of the main results is to show that CH is equivalent to the existence of a weak selection f for which λ f ⁢ ( A ) = 0 {\lambda_{f}(A)=0} whenever | A | ≤ ω {\lvert A\rvert\leq\omega} and λ f ⁢ ( A ) = ∞ {\lambda_{f}(A)=\infty} otherwise. Some conditions are given for a σ-ideal of ℝ {\mathbb{R}} in order to be exactly the family 𝒩 f {\mathcal{N}_{f}} of λ f {\lambda_{f}} -null subsets for some weak selection f. It is shown that there are 2 𝔠 {2^{\mathfrak{c}}} pairwise distinct ideals on ℝ {\mathbb{R}} of the form 𝒩 f {\mathcal{N}_{f}} , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that 𝒩 f {\mathcal{N}_{f}} is exactly the σ-ideal of meager subsets of ℝ {\mathbb{R}} . Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of λ f {\lambda_{f}} -measurable sets.

Universal sets for ideals

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^\omega$ and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for $\mathcal{E}$ - the $\sigma$-ideal generated by closed null subsets of $2^\omega$, and for some ideals connected with forcing notions: $\mathcal{K}_\sigma$ subsets of $\omega^{\omega}$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $\Sigma^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.

Baire theorem for ideals of sets

We study ideals I on N satisfying the following Baire-type property: if X is a complete metric space and{XA:A∈I}is a family of nowhere dense subsets of X withXA⊂XBwheneverA⊂B, then⋃A∈IXA≠X. We give several characterizations and determine the ideals having this property among certain classes like analytic ideals and P-ideals. We also discuss similar covering properties when considering families of compact and meager subsets of X.

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu(U)<\mu(V)$ there is a clopen set $W\subset V$ with $\mu(W)=\mu(U)$. We study $\sigma$-ideals with Borel base on $X$ which are invariant under the action of the group $H_\mu(X)$ of measure-preserving homeomorphisms of $(X,\mu)$, and show that any such $\sigma$-ideal $\mathcal I$ is equal to one of seven $\sigma$-ideals: $\{\emptyset\}$, $[X]^{\le\omega}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal N$, or $[X]^{\le \mathfrak c}$. Here $[X]^{\le\kappa}$ is the ideal consisting of subsets of cardiality $\le\kappa$ in $X$, $\mathcal M$ is the ideal of meager subsets of $X$, $\mathcal N=\{A\subset X:\mu(A)=0\}$ is the ideal of null subsets of $(X,\mu)$, and $\mathcal E$ is the $\sigma$-ideal generated by closed null subsets of $(X,\mu)$.

Non-meager free sets for meager relations on Polish spaces

of nowhere densesubsets of X;• is nowhere meager in X, if for any non-empty open set U⊂ Xthe intersection U∩Ais notmeager in X.It is clear that a subset A⊂ X of a Polish space X is nowhere meager if and only if Ais densein Xand contains no open meager subspace. By definition, cof(M) is the minimal cardinality of acollection X of meager subsets of the Baire space ω

On sequentially closed subsets of the real line in

We show: iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset. The statement “ is sequential” is equivalent to each one of the following propositions: Every sequentially closed subset A of includes a countable cofinal subset C, for every sequentially closed subset A of , is a meager subset of , for every sequentially closed subset A of , , every sequentially closed subset of is separable, every sequentially closed subset of is Cantor complete, every complete subspace of is Cantor complete.

The Structure of the Real Line in Quantum Mechanics and Cosmology

We discuss the recently proposed model, where the spacetime in large scales is parametrized by the usual real line R, while at small (quantum mechanical) scales, the space is parametrized by the real numbers RM from some formal model M of Zermelo–Fraenkel set theory. We argue that the set-theoretic forcing is an important ingredient of the shift from microto macroscale. The set RM , describing the space at the Planck era, is merely a meager subset of R. It is Lebesgue non-measurable and all its measurable subsets have Lebesgue measure 0. According to this, the contributions to the cosmological constant from the zero-point energies of quantum fields vanish. Moreover, the emerged irregularities in the real line can be considered as the source of the primordial quantum fluctuations.

Normal bundles of convex bodies

The normal bundle of an (o-symmetric, smooth and strictly) convex body C in Ed gives rise to a closed convex cone in Ed2, the normal bundle coneNC of C. This article deals with properties of the cone NC and relations between properties of NC and C:The family of normal bundle cones is a meager subset of the space of all closed convex cones in Ed2. To find out whether a cone is a normal bundle cone, a simple criterion is given. The dimension of a normal bundle cone NC is between 12d(d+1) and d2. The lower bound is attained precisely for ellipsoids. As the dimension of NC increases, the ellipsoidal character of the convex body C and the family of linear vector fields which are tangent on the boundary of C decrease. For generic C the dimension of NC is d2. For d=2,3 a complete description of the situation is given. Next, symmetry properties are studied. The cone NC coincides with its polar, its polar transpose, or its transpose, if and only if C is a disc with rotational symmetry of order 4, a Radon disc, a Euclidean ball, or an ellipsoid. A conjecture deals with isometries and linear automorphisms of NC and C. A relation between symmetry properties of a pair of normal bundle cones and orthogonality in two normed spaces is stated. There is a bijection between the family of certain faces of NC and the family of planar shadow boundaries of C with respect to parallel illumination. A conjecture deals with a more general case.

Universal nowhere dense and meager sets in Menger manifolds

In each Menger manifold M we construct:•a closed nowhere dense subset M0 which is homeomorphic to M and is universal nowhere dense in the sense that for each nowhere dense set A⊂M there is a homeomorphism h of M such that h(A)⊂M0;•a meager Fσ-set Σ0⊂M which is universal meager in the sense that for each meager subset B⊂M there is a homeomorphism h of M such that h(B)⊂Σ0. Also we prove that any two universal meager Fσ-sets in M are ambiently homeomorphic.

Dichotomies for C0(X) and C b (X) spaces

Jachymski showed that the set $$\left\{ {(x,y) \in {{\rm{c}}_0} \times {{\rm{c}}_0}:\left( {\sum\limits_{i = 1}^n {\alpha (i)x(i)y(i)} } \right)_{n = 1}^\infty {\rm{ is bounded}}} \right\}$$ is either a meager subset of c 0 × c 0 or is equal to c 0 × c 0. In the paper we generalize this result by considering more general spaces than c 0, namely C 0(X), the space of all continuous functions which vanish at infinity, and C b (X), the space of all continuous bounded functions. Moreover, we replace the meagerness by σ-porosity.

On self-similarities of ergodic flows

Given an ergodic flow $T=(T_t)_{t\in\Bbb R}$, let $I(T)$ be the set of reals $s\ne 0$ for which the flows $(T_{st})_{t\in\Bbb R}$ and $T$ are isomorphic. It is proved that $I(T)$ is a Borel subset of $\Bbb R^*$. It carries a natural Polish group topology which is stronger than the topology induced from $\Bbb R$. There exists a mixing flow $T$ such that $I(T)$ is an uncountable meager subset of $\Bbb R^*$. For a generic flow $T$, the transformations $T_{t_1}$ and $T_{t_2}$ are spectrally disjoint whenever $|t_1|\ne |t_2|$. A generic transformation (i) embeds into a flow $T$ with $I(T)=\{1\}$ and (ii) does not embed into a flow with $I(T)\ne \{1\}$. For each countable multiplicative subgroup $S\subset\Bbb R^*$, it is constructed a Poisson suspension flow $T$ with simple spectrum such that $I(T)=S$. If $S$ is without rational relations then there is a rank-one weakly mixing rigid flow $T$ with $I(T)=S$.

Additivity of the two-dimensional Miller ideal

Let J(M^2) denote the sigma-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of J(M^2) is bigger than the covering number of the ideal of the meager subsets of the Baire space. We also show that Martin's Axiom implies that the additivity of J(M^2) is the cardinality of the continuum. Finally we prove that there are no analytical infinite maximal antichains in any finite product of the power set of the natural numbers modulo the ideal of the finite subsets of the natural numbers.

Metric Baumgartner theorems and universality

It is consistent with the axioms of set theory that for every metric space $X$ which is isometric to some separable Banach space or to Urysohn's universal separable metric space $\mathbb U$ the following holds: \begin{quote} $ {(\star)_X}$ \emph{There exists a nowhere meager subspace of $X$ of cardinality $\aleph_1$ and any two nowhere meager subsets of $X$ of cardinality $\aleph_1$ are almost isometric to each other.} \end{quote} As a corollary, it is consistent that the Continuum Hypothesis fails and the following hold: \begin{enumerate} \item There exists an almost-isometry ultrahomogeneous and universal element in the class of separable metric spaces of size $\aleph_1$. \item For every separable Banach space $X$ there exists an almost-isometry conditionally ultrahomogeneous and universal element in the class of subspaces of $X$ of size $\aleph_1$. \item For every finite dimensional Banach space $X$, there is a \emph{unique} universal element up to almost-isometry in the class of subspaces of $X$ of size $\aleph_1$ \end{enumerate}

Strong mixing Markov semigroups on C1are meager

We show that the set of those Markov semigroups on the Schatten class ${\cal C}_{1}$ such that in the strong operator topology $\lim_{t \to \infty} T(t) = Q $, where $Q$ is a one-dimensional projection, form a meager subset of all Markov semigroups.

A nonlinear Banach–Steinhaus theorem and some meager sets in Banach spaces

We establish a Banach&#8211;Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of $L^1(\mu )\times L^1(\mu )$ and $c_0\time

Baire category and nowhere differentiability for feasible real functions

AbstractA notion of resource‐bounded Baire category is developed for the class PC[0,1] of all polynomial‐time computable real‐valued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resource‐bounded Banach‐Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexity‐theoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Some remarks on totally imperfect sets

We prove the following two theorems. Theorem 1. Let X be a strongly meager subset of 2 ω×ω . Then it is dual Ramsey null. We will say that a σ-ideal I of subsets of 2 satisfies the condition () iff for every X C 2, if ∀ f ∈ ω ↑ ω {g ∈ ω ω : ¬(f? g)} ∩ X ∈ I, then X ∈ I. Theorem 2. The σ-ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition ().