Countable Research Articles

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.

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.

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 [Formula: see text], we apply Ulam’s method with [Formula: see text] 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.

Pengaruh Olahraga Permainan Tradisional Congklak Terhadap Perkembangan Psikomotorik dan Kognitif Siswa Sekolah Dasar

Traditional sports and games, such as congklak, have been an inseparable part of culture and people's lives since ancient times. This research aims to reveal the influence of the congklak game on psychomotor and cognitive development in elementary school students. By comparing groups of students who regularly play congklak and those who don't, this research wants to find out whether this traditional game can improve students' fine motor skills, hand-eye coordination, counting ability, and problem-solving ability. It is hoped that the results of this research can contribute to cultural preservation and improving the quality of education by integrating traditional games as an effective learning medium.

The Effect of Ethnomathematics Approach on Numbers 1-20 Counting Skills for Children 4-6 Years Old

Children aged 0-6 years are at an age where cognitive development is very rapid. Counting ability is one of the things that cannot be ignored in this case. The Ethnomathematics approach is an approach that prioritises cultures that contain mathematical values This study aims to determine the ethnomathematics approach to counting skills 1- 20 students aged 4-6 years. This research is a quasi-experimental research with posttest only control design, with a sample of 22 children out of 44 students at RA Al-Qur'an Dina Padang Matinggi. The results showed a significant effect of initial counting ability on the final counting ability of experimental class students using the ethnomathematics approach.experimental class students who used the ethnomathematics approach. Statistical results show the value of F = 18,861 with a significance of 0.000 (p &lt; 0.05). The Ethnomathematics approach can be used in the learning process at the early childhood education level. So that counting skills can develop through this approach. And of course it is necessary to pay attention to the understanding of the culture possessed by students after the learning process is carried out

The “How many?” task inadequately assesses the understanding of the cardinality principle

Once children have acquired the cardinality principle, they understand that the last number-word used in counting represents the total number of objects in a set. This principle is often assessed using the “How many?” task, which consists in asking “How many?” objects there are in a set after children have counted them. However, we show in this study that out of 188 kindergarteners (mean age: 4 ½ years), 42 (22,3 %) succeeded in repeating the last count word in the “How many?” task but failed to correctly apply the one-to-one correspondence principle during counting. Even when only the easiest countable sets were considered (i.e., linear and homogeneous collections or sets with a very limited number of objects), still more than 10 % of children from our sample repeated the last count word but failed to apply the one-to-one correspondence principle. Such developmental profile, in which children understand that the last word used in counting represents the total number of objects in a set but fail to grasp that each individual object must be associated with a single number word to determine this total, is not psychologically plausible. We conclude that the “How many?” task leads to an inaccurate assessment of the cardinality principle, in both its basic and more meaningful conception, in a non-negligible number of young children.

Coincidence of the Dimensions of First Countable Spaces with a Countable Network

Coincidence of the Dimensions of First Countable Spaces with a Countable Network

Technology-Integrated Instructions and Basic Mathematics Skills Acquisition in Public Secondary Schools in Rwanda: A Case of Nyagatare District

This research investigated the effects of technology-integrated instructions on basic mathematics skills in Rwandan secondary education. The study aimed to assess the types of technology-integrated instructional methods used for acquiring basic math skills in Rwandan public secondary schools, assess the level of mathematics skills and performance in these schools, and determine the relationship between technology-integrated teaching and the acquisition of basic math skills. Schools were selected based on their location and type. Theory used in this research project was cognitive load theory and constructivism theory. The target population included 214 teachers, 422 students, and 80 head teachers, total of 716 respondents, with a sample size of 257 participants. Data was gathered using questionnaires, interviews, and observation, employing purposive, stratified, and simple random sampling methods. Both qualitative and quantitative data collection and analysis methods were used. Qualitative data were analysed through content analysis, while quantitative data were presented using descriptive statistics (frequency, percentage, mean, and standard deviation) and inferential statistics (correlational and regression analysis) with IBM SPSS Version 21.0. Findings for the first objective showed that 85.7% of respondents strongly agreed on the use of computer algebra systems, 93.5% strongly agreed on blended learning and online education, 87.0% strongly agreed on the use of mobile and handheld devices, and 63.6% strongly agreed on software applications for mathematics calculations as methods influencing the acquisition of basic math skills. Regarding the second objective, 87.0% of teachers strongly agreed that verbal counting ability indicates math skill levels, 93.5% agreed that identifying more and less of a quantity shows math skills, 85.7% agreed on problem-solving skills as an indicator, 66.2% on recognizing numerals, and 77.9% on understanding size, shape, and patterns as indicators of math skills and performance. The study revealed a strong positive relationship between skills such as identifying quantities, verbal counting, problem-solving, and blended learning or online education. Problem-solving skills were also positively linked with computer algebra systems and blended learning. The significance of verbal counting was confirmed, as indicated by a p-value of less than 0.05. The study recommends using digital tools like calculators and graphing software to improve understanding of mathematical concepts. Real-world examples and multimedia can foster problem-solving abilities, while group work can enhance teamwork and communication skills. Educational software and online platforms provide personalized learning, enabling students to learn at their own pace and receive specific support.

Prediction of Gestational Diabetes by Second Trimester Complete Blood Count Derived Inflammatory Markers

Abstract Aim: To investigate the predictive ability of routine complete blood count (CBC) derived inflammatory markers on gestational diabetes mellitus (GDM) in the second trimester. Methods: A total of 181 patients, whose routine complete blood count (CBC) was measured between 14 and 24 weeks of gestation, were divided into two groups according to a 75 g OGTT results. 99 women were GDM and 82 women had normal glucose tolerance (NGT). The groups were compared regarding mean platelet volume (MPV), platelet distribution width (PDW), plateletocrit (PCT), immature granulocyte count, delta neutrophil index (DNI), neutrophil/lymphocyte ratio (NLR), platelet/lymphocyte ratio (PLR) and Systemic Immune-Inflammation Index (SII). Results: PDW, immature granulocyte count, NLR, PLR and SII were higher in the GDM group, but no statistically significant difference was observed. Mean values of DNI were lower in women with GDM than NGT, although not statistically significant (0.84 versus 0.99, p =0.742). Conclusion: Our findings revealed that using second trimester inflammatory CBC parameters prior to screening test is not an accurate method to predict GDM.

Distance-aware network for physical-world object distribution estimation and counting

Crowd counting has become popular due to its applications in congested scenes. Current methods excel with specialized datasets but often ignore the density distribution affected by perspective. To investigate the true distribution of crowd density in the physical world, this paper introduces a framework leveraging head size variations to estimate real crowd distribution and count crowds. First, a convolutional neural network is designed to generate predicted density maps and corresponding bounding boxes across different datasets. Second, a filter kernel is employed to identify the most crowded area in the input image based on the obtained boxes. Finally, the vanishing point is computed by calculating the intersection of the two generated lines, which is then used to obtain the inverse perspective of the predicted density map. The Grid Mean Relative Error (GMRE) metric is proposed for evaluating transformation accuracy. In comparison with the Grid Average Mean Absolute Error (GAME), GMRE is distance-aware and more suitable for evaluating differences between distinct coordinate systems. Additionally, extensive experiments are conducted to validate the counting capabilities of the proposed network. Experimental results show the network’s competitive counting ability and reduced transformation error.

Dual discreteness of Σ-products and irreducibility of infinite products

This paper includes two main results. Dual discreteness is a well known generalization of D-spaces. The first one is that every Σ-product of compact metric spaces is dually discrete. The property aD is another generalization of D-spaces, and it implies irreducibility. The second one is that the product Nω1 of ω1 many copies of N is irreducible, where N denotes an infinite countable discrete space.

Weakly periodic gibbs measures for the HC model with a countable set of spin values

Weakly periodic gibbs measures for the HC model with a countable set of spin values

Updating Prospective Self-Efficacy Beliefs About Cardiac Interoception in Anorexia Nervosa: An Experimental and Computational Study.

Patients with anorexia nervosa (AN) typically hold altered beliefs about their body that they struggle to update, including global, prospective beliefs about their ability to know and regulate their body and particularly their interoceptive states. While clinical questionnaire studies have provided ample evidence on the role of such beliefs in the onset, maintenance, and treatment of AN, psychophysical studies have typically focused on perceptual and 'local' beliefs. Across two experiments, we examined how women at the acute AN (N = 86) and post-acute AN state (N = 87), compared to matched healthy controls (N = 180) formed and updated their self-efficacy beliefs retrospectively (Experiment 1) and prospectively (Experiment 2) about their heartbeat counting abilities in an adapted heartbeat counting task. As preregistered, while AN patients did not differ from controls in interoceptive accuracy per se, they hold and maintain 'pessimistic' interoceptive, metacognitive self-efficacy beliefs after performance. Modelling using a simplified computational Bayesian learning framework showed that neither local evidence from performance, nor retrospective beliefs following that performance (that themselves were suboptimally updated) seem to be sufficient to counter and update pessimistic, self-efficacy beliefs in AN. AN patients showed lower learning rates than controls, revealing a tendency to base their posterior beliefs more on prior beliefs rather than prediction errors in both retrospective and prospective belief updating. Further explorations showed that while these differences in both explicit beliefs, and the latent mechanisms of belief updating, were not explained by general cognitive flexibility differences, they were explained by negative mood comorbidity, even after the acute stage of illness.

Roe functors preserve homotopies

We show that coarse maps between countable metric spaces of bounded geometry induce natural transformations of sufficiently good endofunctors of C ∗ C^{*} -algebras and prove that this correspondence is invariant with respect to coarse homotopies.

Sorogan-Based Learning to Stimulate Early Counting Skills

Beginning counting ability is an important basic ability for students to be able to answer and understand further mathematical problems, including the introduction of numbers and basic mathematical concepts. The problem discussed here is an attempt to change the conventional teaching method (lecture) at the beginning counting learning stage towards a learning approach that can better facilitate children's learning needs and create meaningful learning. This study aims to analyze the application of the Sorogan method assisted by graded student worksheets (LKS) in stimulating student’s counting skills in Calistung class. The research method used is descriptive qualitative, with a case study approach. This research was conducted involving Calistung B class students consisting of 5 children as research subjects. The research techniques used were observation, interview, and documentation. The results showed that the Sorogan method assisted by graded worksheets was able to make a positive contribution in stimulating student’s beginning counting skills. Components such as initial apperception, practice problems, discussion, and reflection applied in Sorogan learning activities play an important role in supporting the formation of deep understanding. The student-centered learning process of the Sorogan method, which prioritizes a focus on exploring aspects of individuality, supported by the use of graded worksheets, gradually helps students understand numeracy concepts, thus becoming the foundation for the continuous development of numeracy skills. This method can be an alternative for educators to improve the quality of learning early counting.