Countable Research Articles

Controlling formal fibers of countably many principal prime ideals

Let [Formula: see text] be a complete local (Noetherian) ring. For each [Formula: see text], let [Formula: see text] be a nonempty countable set of nonmaximal, pairwise incomparable prime ideals of [Formula: see text], and suppose that if [Formula: see text], then either [Formula: see text] or no element of [Formula: see text] is contained in an element of [Formula: see text]. We provide necessary and sufficient conditions for [Formula: see text] to be the completion of a local integral domain [Formula: see text] satisfying the condition that, for all [Formula: see text], there is a nonzero prime element [Formula: see text] of [Formula: see text] such that [Formula: see text] is exactly the set of maximal elements of the formal fiber of [Formula: see text] at [Formula: see text]. We then prove related results where the domain [Formula: see text] is required to be countable and/or excellent.

On cycle slipping in infinite-dimensional control systems with periodic nonlinearities

In this paper we consider control systems with periodic nonlinearities characterized by countable sets of equilibria, both Lyapunov stable and unstable. The simplest example of such a system is the mathematical pendulum; therefore, these systems are often called “pendulum-like” systems. In pendulum-like systems, the very concept of stability differs from that in systems with a unique equilibrium. Stability is defined as the convergence of any solution to a certain equilibrium. For stable pendulum-like systems, the problem of cycle slipping arises. In the case of a mathematical pendulum, the number of slipped cycles corresponds to the number of rotations of the pendulum around its suspension point. In general, it represents the distance between the initial value of the input and its limit value. In this paper, we obtain frequency-domain estimates for the number of slipped cycles in infinite-dimensional systems using the Popov method of a priori integral indices. These estimates are tighter than those established in previous works. The paper presents an expanded version of the talk delivered at the International Conference on Physics and Control (PhysCon 2024).

Reflecting topological properties in closed separable subspaces

We establish that, for any metrizable space X, if every closed separable subspace of X has the Baire property, then the space X has the Baire property. The same reflection result holds for spaces Cp(X). It is also shown that all closed separable subspaces of Cp(X) are Čech-complete if and only if X is a functionally countable P-space. If A¯ is pseudocomplete for every countable set A⊂Cp(X), then Cp(X) is pseudocomplete. To show that zero-dimensionality does not reflect in closed separable subspaces of metrizable spaces, under Jensen’s Axiom ♢, we give an example of a connected metrizable space M such that A¯ is countable and hence zero-dimensional for every countable set A⊂M.

Integration of Social Media as A Learning Tool in Mamba’ul ‘Ulum Private Ibtidaiyah School

This study focuses on children's developmental progress at Raudhatul Athfal Al-Bashitiyah concerning learning activities in early childhood education RA Al-Basithiyah, specifically in group B, where the children's numerical recognition skills are notably deficient. When requested, numerical figures should be displayed in the corresponding forms of the digits. Furthermore, only 13 youngsters can write the digits 1-10, while only 17 children can sequentially recite the numbers 1-10. The children's incapacity at RA Al-Basithiyah to recognize and recite the numbers 1-10 is due to the suboptimal teaching method employed for number recognition. The introduction of numbers needs to be supported more by modest learning material, which fails to enhance children's number recognition skills. This study seeks to ascertain if using number cards as a medium can improve children's counting abilities at RA Al Bashitiyah. This study constitutes a Classroom Action Research (PTK). This research employs observation and documentation as data-collecting tools. The participants in this study are the offspring of RA Al Bashitiyah. The study's results utilized two cycles: in the first cycle, the MB category comprised 18 children (60%) and the BSH category included 12 children (40%); in the second cycle, the MB category contained two children (7%) while the BSB category encompassed 28 children (93%), indicating the completion of the second cycle. The number card game enhances children's counting abilities

A Low-Cost Microfluidic Device For the On-Line Counting of Microparticle/Bacteria.

On-line counting of the microparticle/bacteria in the liquid medium has great potential in the food safety and biomedical fields. A new low-cost microfluidic device is proposed for the on-line counting of the microparticles/bacteria in the liquid medium. The gradually contracted microchannel and the viscoelastic fluid are combined to achieve the efficient elastic focusing of the particle/bacteria, which significantly improves the counting accuracy by aligning all particles/bacteria in a single position at the center of the microchannel. A simple light resistance-based counting method is designed and integrated with the microchannel, where the low-cost elements including the laser pointer, convex lens, diaphragm, and photodiode, are used to build the optical path of detection. The influence of the flow rate, the particle size, the property of fluid, and the channel structure on the focusing of the particle/bacteria are investigated by the experiment, and the counting ability of the integrated microfluidic device is validated by using different-sized microparticles and the bifidobacterium. With its simple structure, low cost, easy operation, and high efficiency, this microfluidic device is suitable for the commercial applications, such as the on-line counting of the plastic microparticle in water or the colony-forming units (CFU) of bacteria in food.

Symbolic representations of infinity: the impact of notation and numerical syntax

Past research indicates that concepts of infinity are not fully understood. In countably infinite sets, infinity is presumed to be perceived as larger than any finite natural number. This study explored whether symbolic representations of infinity are processed as such through contrasts with Arabic and verbal written numbers. Comparisons between the infinity word and number words were responded to faster than comparisons of two number words, but not when the infinity symbol was solely compared to Arabic numbers. Moreover, infinity comparisons yielded distance-like effects, suggesting that infinity (both word and symbol) can be misconceived as a “natural number” closer to larger numbers than small ones. These findings demonstrate difficulty perceiving the physically smallest stimulus (∞) as the upper end-value and seem to reflect a limited understanding of symbolic forms of infinity among adults. They further highlight the impact of notation and numerical syntax on how we process symbolic numerical information.

Multistability of recurrent neural networks with general periodic activation functions and unbounded time-varying delays

This paper investigates the multistability of recurrent neural networks (RNNs) with unbounded time-varying delays whose activation functions are general periodic functions. The activation function can be linear, nonlinear, or have multiple corner points, as long as it satisfies Lipschitz continuous condition. According to the characteristics of the parameters of the RNNs and the state space division method, the number of equilibrium points (EPs) of the RNNs is split into three categories, which can be unique, finite, or countable infinite. Some sufficient conditions for determining the number of EPs are presented, the criterion of asymptotically stable EPs is deduced, and the attraction basins of stable EPs are estimated. Furthermore, the multistability results in this paper are an extension of some previous results. The theoretical results are validated using three numerical simulation examples.

Generic classes defined by neighborhoods assignments and stars

In this article we have a couple of mains. The first one is to make a generic exposure on the classes that are obtained by using neighborhood assignments or applying the star operator to a given topological property P. The second one is to perform an analysis on the classes defined in the first part of the work, for the numerability property.It is important to note that in Example 3.13 we present a Hausdorff weakly star countable space which is not feebly Lindelof, which gives a negative response to the Question 3.14 from [1].

A computer scientist’s perspective on approximation of IFS invariant sets and measures with the random iteration algorithm

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

APPROXIMATION THEOREMS THROUGHOUT REVERSE MATHEMATICS

Abstract Reverse Mathematics (RM) is a program in the foundations of mathematics where the aim is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. Generally, the minimal axioms are equivalent to the theorem at hand, assuming a weak logical system called the base theory. Moreover, many theorems are either provable in the base theory or equivalent to one of four logical systems, together called the Big Five. For instance, the Weierstrass approximation theorem, i.e., that a continuous function can be approximated uniformly by a sequence of polynomials, has been classified in RM as being equivalent to weak König’s lemma, the second Big Five system. In this paper, we study approximation theorems for discontinuous functions via Bernstein polynomials from the literature. We obtain many equivalences between the latter and weak König’s lemma. We also show that slight variations of these approximation theorems fall far outside of the Big Five but fit in the recently developed RM of new ‘big’ systems, namely the uncountability of ${\mathbb R}$ , the enumeration principle for countable sets, the pigeon-hole principle for measure, and the Baire category theorem.

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

Regularity of the minmax value and equilibria in multiplayer Blackwell games

AbstractA real-valued function φ that is defined over all Borel sets of a topological space is regular if for every Borel set W, φ(W) is the supremum of φ(C), over all closed sets C that are contained in W, and the infimum of φ(O), over all open sets O that contain W.We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player’s minmax value is regular.We then use the regularity of the minmax value to establish the existence of ε-equilibria in two distinct classes of Blackwell games. One is the class of n-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n − 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

Q*-REGULAR SPACES

In this paper, we introduce and study a new class of generalized regular space is called Q*-regular space which is weaker than regularity. The relationships among strongly rg-regular, g-regular, regular, Q*-regular, almost regular and softly regular spaces are investigated. Some of basic properties and characterizations of Q*-regular spaces in the terms of other separation and countability axioms such as semi-regular, Hausdorff, separable, second countable and Lindelof spaces are obtained.

On the superposition and thinning of generalized counting processes

. In this article, we study the merging and splitting of generalized counting processes (GCPs). First, we study the merging of a finite number of independent GCPs and then extend it to the case of countably infinite. The merged process is observed to be a GCP with increased arrival rates. It is shown that a packet of jumps arrives in the merged process according to the Poisson process. Also, we study two different types of splitting of a GCP. In the first type, we study the splitting of jumps of a GCP where the probability of simultaneous jumps in the split components is negligible. In the second type, we consider the splitting of jumps in which there is a possibility of simultaneous jumps in the split components. It is shown that the split components are GCPs with certain decreased jump rates. Moreover, the independence of split components is established. Later, we discuss applications of the obtained results to industrial fishing problem and hotel booking management system.

COUNTABLE SPACES, REALCOMPACTNESS, AND THE PSEUDOINTERSECTION NUMBER

Abstract All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $ . Our main result shows that, given a cardinal $\kappa $ , the following conditions are equivalent: • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $ . • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$ . In fact, in the case $\mathfrak {d}\leq \kappa \leq \mathfrak {c}$ , every countable dense subspace of $2^\kappa $ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa $ has pseudocharacter at most $\kappa $ in any compactification. This will allow us to calculate $\mathsf {Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space X.

Comparing functional countability and exponential separability

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponential separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the Gδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\delta $$\\end{document}-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

Incidence of myelosuppression in AML is higher compared with that in ALL.

Acute myeloid leukemia (AML) and acute lymphocytic leukemia (ALL) are two subtypes of acute leukemia. However, studies investigating the ability of complete blood count (CBC) parameters to distinguish between patients with AML and ALL remain scarce in the literature. The objective of the present study was to compare the parameters of CBC analysis between Chinese patients with AML and ALL and between patients with M3 AML and non-M3 AML. Prognostic factors for overall survival were also estimated, including sex, age, white blood cell count and hemoglobin. The present study included 147 patients, including children and adults, with newly diagnosed acute leukemia. Information on the age, sex, leukemia subtype, initial CBC results and clinical follow-up findings was recorded and compared between the indicated groups using statistical tests of Mann-Whitney U test and χ2 test. Leukopenia (white blood cell count <3.5x109/l), both leukopenia and anemia, both leukopenia and thrombocytopenia and pancytopenia were found to be significantly more frequent among patients with AML compared with that in patients with ALL (P=0.015, 0.016, 0.015 and 0.019, respectively). For patients with ALL, anemia was recognized as a predictor of a favorable outcome (Hazard ratio, 0.185; 95% CI, 0.046-0.747; P=0.018). These findings suggest that normal hematopoiesis is more frequently inhibited in patients with AML compared with that in patients with ALL. Patients with AL with peripheral blood findings indicative of leukopenia, pancytopenia, or both leukopenia and anemia or both leukopenia and thrombocytopenia are more likely to have AML.

Ulam’s Method for Computing Stationary Densities of Invariant Measures for Piecewise Convex Maps with Countably Infinite Number of Branches

Let [Formula: see text] be a piecewise convex map with countably infinite number of branches. In [ Góra et al. , 2022 ], the existence of Absolutely Continuous Invariant Measure (ACIM) [Formula: see text] for [Formula: see text] and the exactness of the system [Formula: see text] have been proven. In this paper, we develop an Ulam method for approximation of [Formula: see text], the density of ACIM [Formula: see text]. We construct a sequence [Formula: see text] of maps [Formula: see text] s.t. [Formula: see text] has a finite number of branches and the sequence [Formula: see text] converges to [Formula: see text] almost uniformly. Using supremum norms and Lasota–Yorke-type inequalities, we prove the existence of ACIMs [Formula: see text] for [Formula: see text] with the densities [Formula: see text]. For a fixed n, we apply Ulam’s method with k subintervals to [Formula: see text] and compute approximations [Formula: see text] of [Formula: see text]. We prove that [Formula: see text] as [Formula: see text] both a.e. and in [Formula: see text]. We provide examples of piecewise convex maps [Formula: see text] with countably infinite number of branches and their approximations by [Formula: see text]’s with finite number of branches. For the increasing values of parameter [Formula: see text] we calculate the errors [Formula: see text].

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

MA and three Fréchet spaces

M. Scheepers introduced the notion of selectively separable as one of many interesting selection principles. Fréchet spaces are known to be selectively separable but neither of these properties is well-behaved in products, even for countable spaces. We prove that the open coloring axiom, OCA, implies that the product of two countable Fréchet spaces is selectively separable. We prove that Martin's Axiom implies there are three countable Fréchet spaces whose product is not selectively separable.