Convergence Of Sequence Research Articles

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.

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.

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.

Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on R n \mathbb {R}^n . While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on R n \mathbb {R}^n .

Exploring Iterated Implicit Function Systems: Existence and Properties of Attractors

This paper investigates a type of iterated implicit function systems composed of equations [Formula: see text], where [Formula: see text] is a continuous function, and [Formula: see text] is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation [Formula: see text] containing the implicit function or being [Formula: see text]-contractive about [Formula: see text]. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.

The Josefson–Nissenzweig theorem and filters on $$\omega $$

AbstractFor a free filter F on $$\omega $$ ω , endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F } , where $$p_F\not \in \omega $$ p F ∉ ω , with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F . Spaces of the form $$N_F$$ N F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z , that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z . Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z , then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

Accelerated numerical solutions for discretized Black–Scholes equations

Abstract Accepted by: Aris Syntetos This study thoroughly investigates the efficiency of advanced numerical extrapolation methods aimed at enhancing the convergence of vector sequences in the realm of mathematical finance. Our focus lies in the application of polynomial extrapolation techniques to calculate finite difference solutions for the Black–Scholes (BS) equation—an indispensable model in options pricing. The performance of our algorithms undergoes rigorous evaluation through a comprehensive analysis involving both simulated and real-world data. Notably, our experiments uncover that a stochastic scheme, incorporating two extrapolation strategies and a random relaxation parameter, outperforms other proposed methods, excelling in both convergence and stability metrics. Our findings underscore the potential of this numerical extrapolation method to enhance the efficiency of financial calculations, particularly in the realm of option pricing. This innovation holds promise for refining financial models and addressing specific challenges within the field of mathematical programming, providing effective solutions to the primary computational bottlenecks commonly encountered in financial decision-making.

Transitivity in nonautonomous systems generated by a uniformly convergent sequence of maps

Let (X,d) be a metric space and f1,∞={fn}i=0∞ be a sequence of continuous maps fn:X→X such that (fn) converges uniformly to a continuous map f. We investigate which conditions ensure that the transitivity of functions fn or the transitivity of the nonautonomous system (X,f1,∞) is inherited to the limit function f and vice versa. Such problem have been studied for instance by A. Fedeli, A. Le Donne or J. Li who give different sufficient condition for inheriting of transitivity from fn to f. In this paper we give a survey of known result relating to this problem and prove new results concerning transitivity.

Hierarchical temporal sequence convergence in prescribed-time control for quadrotor UAVs under unknown dynamic disturbances

This paper studies the trajectory tracking control problem for quadrotor unmanned aerial vehicles (UAVs) under unknown disturbances and model uncertainties. A priori information about disturbances is required for most existing prescribed time-extended state observer (PTESO) methods. For unknown disturbances, a barrier function-based adaptive prescribed-time extended state observer (BFAPTESO) is designed by utilizing the time-domain transformation and adaptive mechanism in the prescribed time and using the barrier function method in the rest time. To improve tracking performance amidst model uncertainty, a finite-gain prescribed-time state observer (FGPTSO) is applied in the position loop. Then, the novel prescribed-time controllers are designed in conjunction with the above observers in the attitude and position loops. The proposed control scheme mitigates unknown dynamic disturbances and model uncertainties, guaranteeing that the tracking error achieves and sustains the preset accuracy within the prescribed time. Finally, numerical results indicate that the proposed control method is effective and robust.

Accelerated Value Iteration for Nonlinear Zero-Sum Games with Convergence Guarantee

In this paper, an accelerated value iteration (VI) algorithm is established to solve the zero-sum game problem with convergence guarantee. First, inspired by the successive over relaxation theory, the convergence rate of the iterative value function sequence is accelerated significantly with the relaxation factor. Second, the convergence and monotonicity of the value function sequence are analyzed under different ranges of the relaxation factor. Third, two practical approaches, namely the integrated scheme and the relaxation function, are introduced into the accelerated VI algorithm to guarantee the convergence of the iterative value function sequence for zero-sum games. The integrated scheme consists of the accelerated stage and the convergence stage, and the relaxation function can adjust the value of the relaxation factor. Finally, including the autopilot controller, the fantastic performance of the accelerated VI algorithm is verified through two examples with practical physical backgrounds.

Dollo Parsimony Overestimates Ancestral Gene Content Reconstructions.

Ancestral reconstruction is a widely used technique that has been applied to understand the evolutionary history of gain and loss of gene families. Ancestral gene content can be reconstructed via different phylogenetic methods, but many current and previous studies employ Dollo parsimony. We hypothesize that Dollo parsimony is not appropriate for ancestral gene content reconstruction inferences based on sequence homology, as Dollo parsimony is derived from the assumption that a complex character cannot be regained. This premise does not accurately model molecular sequence evolution, in which false orthology can result from sequence convergence or lateral gene transfer. The aim of this study is to test Dollo parsimony's suitability for ancestral gene content reconstruction and to compare its inferences with a maximum likelihood-based approach that allows a gene family to be gained more than once within a tree. We first compared the performance of the two approaches on a series of artificial data sets each of 5,000 genes that were simulated according to a spectrum of evolutionary rates without gene gain or loss, so that inferred deviations from the true gene count would arise only from errors in orthology inference and ancestral reconstruction. Next, we reconstructed protein domain evolution on a phylogeny representing known eukaryotic diversity. We observed that Dollo parsimony produced numerous ancestral gene content overestimations, especially at nodes closer to the root of the tree. These observations led us to the conclusion that, confirming our hypothesis, Dollo parsimony is not an appropriate method for ancestral reconstruction studies based on sequence homology.

Measure on a Lattice of Infinite Tuples of Real Numbers

We consider the analogue of measure on a ring of sets, by defining a lattice-measure on a lattice-ring of elements of a lattice of infinite tuples of real numbers. We obtain various results about the convergence of sequences of tuples in the lattice with respect to the lattice-measure and use these results to show that the limit of a convergent sequence in a lattice sigma-ring is also in the lattice sigma-ring under certain conditions.

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

A new proximal heavy ball inexact line-search algorithm

AbstractWe study a novel inertial proximal-gradient method for composite optimization. The proposed method alternates between a variable metric proximal-gradient iteration with momentum and an Armijo-like linesearch based on the sufficient decrease of a suitable merit function. The linesearch procedure allows for a major flexibility on the choice of the algorithm parameters. We prove the convergence of the iterates sequence towards a stationary point of the problem, in a Kurdyka–Łojasiewicz framework. Numerical experiments on a variety of convex and nonconvex problems highlight the superiority of our proposal with respect to several standard methods, especially when the inertial parameter is selected by mimicking the Conjugate Gradient updating rule.

Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

Neural Q-learning for discrete-time nonlinear zero-sum games with adjustable convergence rate

In this paper, an adjustable Q-learning scheme is developed to solve the discrete-time nonlinear zero-sum game problem, which can accelerate the convergence rate of the iterative Q-function sequence. First, the monotonicity and convergence of the iterative Q-function sequence are analyzed under some conditions. Moreover, by employing neural networks, the model-free tracking control problem can be overcome for zero-sum games. Second, two practical algorithms are designed to guarantee the convergence with accelerated learning. In one algorithm, an adjustable acceleration phase is added to the iteration process of Q-learning, which can be adaptively terminated with convergence guarantee. In another algorithm, a novel acceleration function is developed, which can adjust the relaxation factor to ensure the convergence. Finally, through a simulation example with the practical physical background, the fantastic performance of the developed algorithm is demonstrated with neural networks.

Convergence of power sequences of B-operators with applications to stability

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2 × 2 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal ( 2 , 2 ) (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed.