Monotone Sequences Research Articles

WEIGHTED ESTIMATE OF A MATRIX OPERATOR WITH VARIABLE UPPER LIMIT ON THE CONE OF MONOTONE SEQUENCES

Hardy's inequality was formulated in 1920 and finally proved in 1925. Since then, this inequality has been significantly developed. The first development was related to the consideration of more general weights. The next step was to use more general operators with other kernels instead of the Hardy operator. Currently, there are many works devoted to Hardy-type inequalities with iterated operators. Motivated by important applications, all these generalizations of Hardy's inequality are studied not only on the cone of non-negative functions, but also on the cone of monotone functions. In this paper, we consider the problem of finding necessary and sufficient conditions for the fulfillment of a weighted Hardy-type inequality on the cone of monotone sequences for 1<p≤q<∞. The main method for solving the problem is the reduction method, which, using the Sawyer principle, allows us to reduce a Hardy-type inequality on the cone of monotone sequences to some inequality for all non-negative sequences.

General bounds for the deficit distribution at ruin in the Sparre Andersen model

We present constructive methods that give upper and lower bounds of Ψ, the insurer’s deficit distribution at ruin time in the Sparre Andersen model. The methodology effectively provides monotone sequences of upper and lower bounds of Ψ, and those sequences converge to Ψ with exponential rate. Several new lower and upper bounds of Ψ are given. As an example, we introduce new bounds of the deficit distribution at ruin when the claim distribution satisfies the new better/worse than used property. We show that some of the examined lower and upper bounds of Ψ may coincide with the exact values of the considered probability.

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

Generalized Fejér monotone sequences and their finitary content

We provide quantitative and abstract strong convergence results for sequences from a compact metric space satisfying a certain form of generalized Fejér monotonicity where (1) the metric can be replaced by a much more general type of function measuring distances (including, in particular, certain Bregman distances), (2) full Fejér monotonicity is relaxed to a partial variant and (3) the distance functions are allowed to vary along the iteration. For such sequences, the paper provides explicit and effective rates of metastability and even rates of convergence, the latter under a regularity assumption that generalizes the notion of metric regularity introduced by Kohlenbach, López-Acedo and Nicolae, itself an abstract generalization of many regularity notions from the literature. In the second part of the paper, we apply the abstract quantitative results established in the first part to two algorithms: one algorithm for approximating zeros of maximally monotone and maximally ρ-comonotone operators in Hilbert spaces (in the sense of Combettes and Pennanen as well as Bauschke, Moursi and Wang) that incorporates inertia terms every other term and another algorithm for approximating zeros of monotone operators in Banach spaces (in the sense of Browder) that is only Fejér monotone w.r.t. a certain Bregman distance.

A new perspective for potassium intercalation chemistry in graphitic carbon materials

Potassium intercalation mechanism in graphitic carbon materials is known to be a staging reaction similar to that of lithium, despite their antithetic intercalation trend in turbostratic carbon (TBC) materials. This study clarified the distinctive potassium intercalation behavior of graphitic carbon materials with different local microstructures through a systematic comparative investigation. In contrast to the monotonic stacking sequence of lithium-intercalated graphitic carbon materials, multiple potassium-intercalated graphitic configurations can be formed by potassiation at an energy level similar to the theoretical KC8 formation. Accordingly, potassium intercalation in TBC materials can result in a theoretical low-voltage plateau capacity without long-range ordering. In contrast, the high energy cost for the threshold number of initial potassium insertions significantly hinders two-phase potassium intercalation in well-ordered graphite-like carbon materials, leading to a poor plateau capacity. Hence, potassium intercalation is more favorable in a turbostratic microstructure that includes intrinsic defects, whereas the presence of too many intrinsic defects reduces the polyhexagonal carbon plane for turbostratic intercalation, leading mainly to solid-solution potassium intercalation. Based on these results, the potassium intercalation mechanism in graphitic carbon materials can be classified into three different types such as 1) solid-solution intercalation, 2) turbostratic intercalation, and 3) two-phase intercalation.

Strictly monotone sequences of lower and upper bounds on Perron values and their combinatorial applications

In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonnegative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.

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.

An Analysis of Sequence and Series

This article provides a comprehensive exploration of sequences and series, fundamental concepts in mathematics with wide-ranging applications. Starting with an introduction to sequences, including arithmetic and geometric sequences, the discussion extends to finite sequences and n-tuple sequences. Arithmetic and geometric series are explored, with formulas for their sums. The article delves into the properties of sequences and series, discussing convergence, divergence, and absolute convergence. Various types of sequences, such as bounded and monotonic sequences, are defined, and the Cauchy Criterion is introduced as a criterion for the existence of the limit of a sequence. Elementary facts about series, including absolute convergence and the number e as the sum of a series, are covered. Additionally, the concept of sequences of functions is introduced. The article concludes by emphasizing the significance of sequences and series in fields like higher mathematics, art, science, technology, and finance, showcasing their crucial role in decision-making, financial analysis, and risk assessment across diverse disciplines.

Existence results for a class of four point nonlinear singular BVP arising in thermal explosion in a spherical vessel

In this article, the following class of four-point singular non-linear boundary value problem (NLBVP) is considered which arises in thermal explosion in a spherical vessel −(s2y′(s))′=s2f(s,y,s2y′),s∈(0,1),y′(0)=0,y(1)=δ1y(η1)+δ2y(η2), where Ω=(0,1)×R2, f:Ω→R is continuous on Ω as well as satisfy Lipschitz condition with respect to y and y′ (one sided), δ1, δ2>0 are constants, and 0<η1≤η2<1. We provide an estimation of the region of existence of a solution of above singular NLBVP. We extend the theory of monotone iterative technique (MIT) which provides computable monotone sequences that converge to the solutions of the nonlinear four point BVPs.

A General Result on (φ, δ)-Monotone Sequences

A General Result on (φ, δ)-Monotone Sequences

Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales

&lt;p&gt;In this study, we introduced several novel Hardy-type inequalities with negative parameters for monotone functions within the framework of delta calculus on time scales $ \mathbb{T} $. As an application, when $ \mathbb{T = N}_{0}, $ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $ \mathbb{T = R}, $ we established continuous analogues of inequalities that have appeared in previous literature. Additionally, we presented inequalities for other time scales, such as $ \mathbb{T} = q^{\mathbb{N}_{0}} $ for $ q &amp;gt; 1, $ which, to the best of the authors' knowledge, represented largely novel contributions.&lt;/p&gt;

Bisymmetric non-negative Jacobi matrix realizations

Within the symmetric inverse eigenvalue problem, the case of bisymmetric Jacobi matrices occupies a central place, since for any strictly monotone list of n real numbers there exists a unique bisymmetric Jacobi matrix realizing the list. Apart from their meaning in several issues such as physics, mechanics, statistics, to cite some of them, the families of this kind of matrices whose spectrum is known are used as models for testing the different algorithms to recover the entries of matrices from spectra data. However, the spectrum is known only for a few families of bisymmetric Jacobi matrices and the examples mainly refer to the case when the spectrum is given by a linear or quadratic function of the order and of the row index. In the first part of this paper, we join all known cases by proving a general result about bisymmetric Jacobi realizations of strictly monotone sequences that are quadratic at most. In the second part, we focus on the non-negative bisymmetric realizations, obtaining new necessary conditions for a given list to be realized by a non-negative bisymmetric Jacobi matrix. The main novelty in our techniques is considering the gaps between the eigenvalues instead of focusing on the eigenvalues themselves. In the last part of this paper, we explicitly obtain the bisymmetric realization of any list for order less or equal to 6.

Monotoniškųjų sekų metodo taikymas

The method of monotonic sequences (see [1], [2]) is considered and the issue of its exactness is discussed. It is shown how the method can be applied to finite sums with a large number of summands.

3D Koch-type crystals

We consider the construction of a family \{K_N\} of 3 -dimensional Koch-type surfaces, with a corresponding family of 3 -dimensional Koch-type “snowflake analogues” \{\mathcal{C}_N\} , where N&gt;1 are integers with N \not\equiv 0 \mathrm{mod}\,\,3 . We first establish that the Koch surfaces K_N are s_N -sets with respect to the s_N -dimensional Hausdorff measure, for s_N=\log(N^2+2)/\log(N) the Hausdorff dimension of each Koch-type surface K_N . Using self-similarity, one deduces that the same result holds for each Koch-type crystal \mathcal{C}_N . We then develop lower and upper approximation monotonic sequences converging to the s_N -dimensional Hausdorff measure on each Koch-type surface K_N , and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set \mathcal{C}_N . As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals \mathcal{C}_N , for N&gt;2 .

Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations.

$$\boldsymbol{\alpha}$$-Monotone Sequences and the Lorentz Theorem

$$\boldsymbol{\alpha}$$-Monotone Sequences and the Lorentz Theorem

Certain aspects of ℐ-statistical supremum and ℐ-statistical infimum of real-valued sequences

Abstract Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ {\mathcal{I}} -statistical supremum and ℐ {\mathcal{I}} -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ {\mathcal{I}} -statistical monotonicity and establish the condition under which an ℐ {\mathcal{I}} -statistical monotonic sequence is ℐ {\mathcal{I}} -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ {\mathcal{I}} -statistical convergence of a real-valued sequence.

A multiscale model for multiaxial inelastic behavior of elastomeric particulate composites

This article addresses the problem of the multiscale constitutive representation of the multiaxial inelastic behavior of elastomeric particulate composites. A fully three-dimensional model is proposed within a micromechanical treatment to describe the multiaxial inelastic response in relation to the reinforcement mechanisms. A network decomposition based on the tube confinement theory is used to consider the combined effects of multiaxiality and inelasticity. The near-field direct interactions between the particles and the rubber networks are physically described using the Eshelby inclusion theory. The capabilities of the microstructure-based model are evaluated for different modes of deformation over a wide range of filler concentrations. It is found being able to successfully reproduce the significant features of the multiaxial macro-response upon monotonic and cyclic loading sequences. Important insights about the effective role of the reinforcement mechanisms on the multiaxial dissipation are revealed.

Monotonicity of the Cores of Massive Stars

Massive stars are linked to diverse astronomical processes and objects including star formation, supernovae and their remnants, cosmic rays, interstellar media, and galaxy evolution. Understanding their properties is of primary importance for modern astronomy, and finding simple rules that characterize them is especially useful. However, theoretical simulations have not yet realized such relations, instead finding that the late evolutionary phases are significantly affected by a complicated interplay between nuclear reactions, chemical mixing, and neutrino radiation, leading to nonmonotonic initial-mass dependencies of the iron core mass and the compactness parameter. We conduct a set of stellar evolution simulations, in which evolutions of He star models are followed until their central densities uniformly reach 1010 g cm−3, and analyze their final structures as well as their evolutionary properties, including the lifetime, surface radius change, and presumable fates after core collapse. Based on the homogeneous data set, we have found that monotonicity is inherent in the cores of massive stars. We show that not only the density, entropy, and chemical distributions, but also their lifetimes and explosion properties such as the proto-neutron-star mass and the explosion energy can be simultaneously ordered into a monotonic sequence. This monotonicity can be regarded as an empirical principle that characterizes the cores of massive stars.

Characterizations of functions of bounded variation on effect algebras

In this paper, the concept of [Formula: see text]-null-additivity on effect algebras is introduced and investigated. Theorems connecting null-additivity and [Formula: see text]-null-additivity are also proved. Further, pseudo-metric generating property is studied and its relation with null–null-additivity is also explored. Some theorems are also obtained as characterizations for functions of bounded variation on effect algebras. Finally, it is shown that limit of every monotone sequence exists for a function of bounded variation defined on effect algebras even if the function is not continuous from above and (or) below.