Convergence Of Sequence Research Articles

Some results on $I_2$-deferred statistically convergent double sequences in fuzzy normed spaces

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-deferred statistical convergence in FNS for double sequences. We further define $I_2$-deferred statistical limit points and $I_2$-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts.

Failure Diagnosis for Dental Air Turbine Handpiece with Payload Using Feature Engineering and Temporal Convolution Network.

The internal mechanisms of dental air turbine handpieces (DATHs) have become increasingly intricate over time. To enhance the operational reliability of dental procedures and guarantee patient safety, this study formulated temporal convolution network (TCN) prediction models with the functions of causality in time sequence, transmitting memory, learning, storing, and fast convergence for monitoring the health and diagnosing the rotor and collet failure of DATHs. A handpiece mimicking a dentist's hand load of 100 g was employed to repeatedly mill a glass porcelain block back and forth for cutting. An accelerometer was employed to capture vibration signals during free-running of unrestrained operation of the handpiece, aiming to discern the characteristic features of these vibrations. These data were then utilized to create a diagnostic health classification (DHC) for further developing a TCN, a 1D convolutional neural network (CNN), and long short-term memory (LSTM) prediction models. The three frameworks were used and compared for machine learning to establish DHC prediction models for the DATH. The experimental results indicate that, in terms of DHC predicted for the experimental dataset, the square categorical cross-entropy loss function error of the TCN framework was generally lower than that of the 1D CNN, which did not have a memory framework or the drawback of the vanishing gradient problem. In addition, the TCN framework outperformed the LSTM model, which required a longer history to provide sufficient diagnostic ability. Still, high accuracies were achieved both in the direction of feed-drive milling and in the gravity of the handpiece through vibration signals. In general, the failure classification prediction model could accurately predict the health and failure mode of the dental handpiece before the use of the DATH when an embedded sensor was available. Therefore, this model could prove to be a beneficial tool for predicting the deterioration patterns of real dental handpieces in their remaining useful life.

Characteristics of pre-service mathematics teacher when solving convergent sequence problems

[Abstract]: Convergent sequences pose a challenge for students to comprehend in real analysis courses. Problem-solving based learning can serve as an alternative approach for imparting the understanding of convergent sequences. The objective of this research is to provide a description of the activities undertaken by students when solving problems related to convergent sequences, focusing on specific characteristics of their problem-solving approaches. This study used a qualitative methodology, namely a case study design, with a sample size of 14 participants who were recruited using purposive sampling. The collection of data was conducted via tests and interviews. The findings of this study indicate that the participants can be classified into dominant representation groups and non-dominant representation groups. The respondents who are verbally dominant tend to express ideas using precise language, possess the ability to elaborate on concepts, and demonstrate logical reasoning and argumentation skills. The respondents who are visually dominant have a tendency to analyze or convert visual representations throughout the process of problem-solving. Meanwhile, those who are symbolically dominant tend to approach problem-solving by breaking down the difficulties into multiple solution phases that are conceptually organized. The respondents who are not dominant in a certain type of representation show flexibility in understanding the problem by using a variety of representations that are appropriate for their situation and level of knowledge. The results of this study can serve as a guide for constructing educational approaches, taking into account the characteristics of students when solving problems related to convergent sequences. [Abstrak]: Barisan konvergen merupakan salah satu konsep yang sulit dipahami peserta didik pada matakuliah analisis real. Pembelajaran berbasis pemecahan masalah dapat menjadi alternatif untuk menanamkan konsep barisan konvergen. Penelitian ini bertujuan untuk mendeskripsikan aktivitas peserta didik dalam memecahkan masalah barisan konvergen berdasarkan representasi matematis yang mereka kembangkan. Kajian ini menggunakan pendekatan kualitatif dengan jenis studi kasus yang melibatkan 14 responden yang dipilih secara purposive sampling. Data dikumpulkan menggunakan instrumen tes dan pedoman wawancara. Hasil penelitian ini menunjukkan bahwa responden dapat dikategorikan menjadi kelompok representasi dominan dan kelompok representasi tidak dominan. Responden yang dominan verbal cenderung menyampaikan ide melalui kata-kata yang jelas, mampu merincikan konsep serta menyusunnya dengan argumen dan pemikiran yang logis. Responden yang dominan visual cenderung menginterpretasikan atau menerjemahkan tampilan visual dalam membangun tahapan pemecahan masalah. Adapun responden dominan simbolik cenderung memecahkan masalah dengan menguraikannya menjadi beberapa langkah penyelesaian yang terstruktur secara konseptual. Karakteristik responden yang tidak dominan pada satu tipe representasi menunjukkan fleksibilitas mahasiswa dalam memahami masalah dengan menggunakan beragam representasi yang disesuaikan dengan kondisi dan pengetahuan yang dimiliki. Temuan penelitian ini dapat menjadi acuan untuk merancang skenario pembelajaran berdasarkan karakteristik individu dalam pemecahan masalah barisan konvergen.

On relative uniform convergence of triple sequence of functions

This paper discusses relative uniform convergence of triple sequence of functions that are defined on a compact domain. Another central idea that is discussed is the regular relative uniform convergence and the Cauchy relative uniform convergence of triple sequence of functions. The idea that a continuous triple sequence defined on a compact domain is relative uniform convergent if and only if it is relative uniform Cauchy has been discussed and established. Subsequently, a glimpse into Cesaro summability of triple sequences and a theorem regarding triple Cesaro summability of bounded relative uniform triple sequence of functions have been introduced.

Anti $ T_{2} $-generalized topological spaces

In this paper, we investigated non strong hyperconnected generalized topological spaces. Ekici \cite{Eki11} and Devi \cite{Ren12} have provided the results of hyperconnectedness for strong generalized topological spaces. We generalized these results for arbitrary generalized topological spaces. Through the notion of hyperconnectedness of arbitrary generalized topological spaces, we constructed an example which fails Hausdorff characterization of topological spaces \lq\lq A first countable spaces is Hausdorff if and only if every convergent sequence has unique limit\rq\rq. This example also serves the purpose of constructing Anti Hausdorff Fr$\acute{e}$chet space in which every convergent sequence has unique limit required by Novak in \cite{Nov39}.

Applications of fractional difference operators for new version of Brudno-Mazur Orlicz bounded consistency theorem

In this paper, we intend to prove that the modulus $\mathcal{A}-$lacunary statistical convergence of fractional difference double sequences and modulus lacunary fractional matrix of four-dimensions taken over the space of modulus $\mathcal{A}-$lacunary fractional difference uniformly integrable real sequences are equivalent. We represent another version of the Brudno-Mazur Orlicz bounded consistency theorem by using modulus function, lacunary sequence, and fractional difference operator. We show that the four-dimensional $RH-$ regular matrices $\mathcal{A}$ and $\mathcal{B}$ are modulus lacunary fractional difference consistent over the multipliers space of modulus fractional difference $\mathcal{A}-$summable sequences and an algebra $Z.$

On lacunary Δ^m-statistical convergence of triple sequence in intuitionisticfuzzy normed space

In this study, we define lacunary Δm-statistical convergence in the framework of intuitionistic fuzzy normed spaces (IFNS) for triple sequences. We prove several results for lacunary Δm-statistical convergence of triple sequence in IFNS. We further established lacunary Δm-statistical Cauchy sequences and provided the Cauchy convergence criterion for this novel idea of convergence.

Some observations on ℐ-statistically pre-Cauchy sequences of complex uncertain variables defined by Orlicz functions

Abstract In this research article, we introduce ℐ {\mathcal{I}} -statistically pre-Cauchy sequences of complex uncertain variables in five different aspects of uncertainty, namely: in mean, in measure, in distribution, in almost sure, and in uniformly almost sure. We also explore the connection between ℐ {\mathcal{I}} -statistically pre-Cauchy sequences and ℐ {\mathcal{I}} -statistically convergent sequences using complex uncertain variables. Additionally, we initiate the study of ℐ {\mathcal{I}} -statistically pre-Cauchy sequences of complex uncertain variables through Orlicz functions.

Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

Convergent sequences in iterated ultrapowers as p-compact groups

We prove, in ZFC, that if G is an infinite countable Abelian group, then there is an ultrafilter p∈ω⁎ such that (Ultp(G),τBohr‾) has non-trivial convergent sequences, consequently (Ultpω1(G),τBohr‾) has non-trivial convergent sequences, extending Theorem 3.9 from [13]. In addition, we prove that the Remark 3.8 from [13] is false; so, the proof of the Corollary 3.11 is false too.

On two consequences of CH established by Sierpiński

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

Hausdorff limits of external rays: The topological picture

AbstractWe study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree with connected Julia sets.

On Common Fixed Point for Contractive Mappings in $p$-Pompeiu-Hausdorff Metric Spaces

In this paper, we establish the existence of a common fixed point from a pair of set-valued mappings. By utilizing the concept of convergence of set-valued mappings' sequences, both ordinary and pointwise convergence, we establish a common fixed point theorem. This our newly result is a generalization of common fixed point theorem of set-valued mappings on partial metric spaces. Further, we establish newly common fixed point theorem under $\phi$-contraction on partial metric spaces.

µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.

On deferred f-statistical convergence for double sequences

Abstract In this article, we first put forward the concept of deferred f-double natural density for double sequences, where f is an unbounded modulus. Then, we combine f-density with deferred statistical convergence for double sequences and investigate deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f. Moreover, we extend these concepts to deferred f-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f-statistical convergence of order α \alpha and strongly deferred f-summability of order α \alpha for double sequences and obtain some conclusions.

Improved multi-scale fusion network for solving non-smooth elliptic interface problems with applications

The utilization of deep learning methodologies for addressing partial differential equations (PDEs) has garnered significant attention in recent years. This paper introduces an improved network structure tailored for the discontinuity-capturing, enabling the resolution of interface problem through a unified neural network framework. Employing the probability space filling argument, we show that our model can generate convergent sequences, where the convergence rate depends on the number of sampling points. Several numerical experiments with regular and irregular interfaces are conducted to elucidate the convergence characteristics, thereby validating the theoretical assertions. Furthermore, we apply our approach to effectively solve the size-modified Poisson-Boltzmann test model, utilizing it for predicting electrostatics and the solvation free energies for proteins immersed in ionic solvents, thus showcasing practical applications of our method.

Wijsman Ideal Convergence Sequence in Neutrosophic Metric Spaces

Objectives: An examination of convergence for a sequence of closed sets in Neutrosophic Metric Spaces is the main goal of this research. Particularly, the focus of this study is directed at Wijsman -convergence ( and Wijsman -convergence . Methods: The various properties of Neutrosophic Metric Spaces are investigated in this study. Using exact mathematical techniques, this study comprehends the features of convergent sequences in these domains. Furthermore, this study expresses particular views on Wijsman -Cauchy and Wijsman -Cauchy sequences. Findings: The convergence behaviour of closed sets in Neutrosophic Metric Spaces is a result of this research. it presents findings and remarks about the sequences of Wijsman - convergence, Wijsman - convergence, Wijsman -Cauchy, and Wijsman -Cauchy. These findings improve the current understanding of convergence in relation to NMS. Novelty: What makes this work unique is that it focuses on convergence notions in Neutrosophic Metric Spaces, specifically on Wijsman and Wijsman - convergence enriches the current body of information in this domain. Keywords: Neutrosophic Metric Spaces, Wijsman I ­convergent and Wijsman I ¿ ­ convergent sequences, Wijsman I ¿ ­Cauchy and Wijman I ­Cauchy sequences

Statistically convergent difference sequences of bi-complex numbers

Abstract In this article we introduce the notion of statistical convergence difference sequences of bi-complex numbers. Some properties of these sequence spaces like BC-module, Banach BC-module, BC-balanced set, BC-convex set, solidness, and symetricity are studied.

Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces

Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces

On ρ-Statistically Convergence Defined by a Modulus Function in Fuzzy Difference Sequences

In this study, we first introduced the definition ∆_ρ^m-statistical convergence for sequences of fuzzy numbers using the generalized difference operator ∆^m. Furthermore, we defined the strong N_F^ρ (∆^m,q)-summable sequence set and the strong N_F^ρ (∆^m,f,q)-summable sequence set for fuzzy difference sequences aided by a modulus function f. Subsequently, we provided certain inclusion theorems between these sets and the S_F^ρ (∆^m ) set.