Foundations

11 pages, 289 KiB

Open AccessArticle

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

by Ioannis K. Argyros, Jai Prakash Jaiswal, Akanksha Saxena and Michael I. Argyros

Foundations 2022, 2(2), 512-522; https://doi.org/10.3390/foundations2020034 - 20 Jun 2022

Viewed by 1254

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a [...] Read more.

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a more interesting case. A process is developed based on majorizing sequences and the notion of restricted Lipschitz condition to provide a semi-local convergence analysis for the third convergent order Noor–Waseem method. Due to the generality of our technique, it can be used on other high order methods. The convergence analysis is enhanced. Numerical applications complete are used to test the convergence criteria. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

8 pages, 248 KiB

Open AccessArticle

Extended Newton–Kantorovich Theorem for Solving Nonlinear Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Foundations 2022, 2(2), 504-511; https://doi.org/10.3390/foundations2020033 - 8 Jun 2022

Viewed by 1529

Abstract

The Newton–Kantorovich theorem for solving Banach space-valued equations is a very important tool in nonlinear functional analysis. Several versions of this theorem have been given by Adley, Argyros, Ciarlet, Ezquerro, Kantorovich, Potra, Proinov, Wang, et al. This result, e.g., establishes the existence and [...] Read more.

The Newton–Kantorovich theorem for solving Banach space-valued equations is a very important tool in nonlinear functional analysis. Several versions of this theorem have been given by Adley, Argyros, Ciarlet, Ezquerro, Kantorovich, Potra, Proinov, Wang, et al. This result, e.g., establishes the existence and uniqueness of the solution. Moreover, the Newton sequence converges to the solution under certain conditions of the initial data. However, the convergence region in all of these approaches is small in general; the error bounds on the distances involved are pessimistic, and information about the location of the solutions appears improvable. The novelty of our study lies in the fact that, motivated by optimization concerns, we address all of these. In particular, we introduce a technique that extends the convergence region; provides weaker sufficient semi-local convergence criteria; offers tighter error bounds on the distances involved and more precise information on the location of the solution. These advantages are achieved without additional conditions. This technique can be used to extend other iterative methods along the same lines. Numerical experiments illustrate the theoretical results. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

16 pages, 479 KiB

Open AccessArticle

Statistical Properties of Estimators of the RMSD Item Fit Statistic

by Alexander Robitzsch

Foundations 2022, 2(2), 488-503; https://doi.org/10.3390/foundations2020032 - 6 Jun 2022

Cited by 6 | Viewed by 2082

Abstract

In this article, statistical properties of the root mean square deviation (RMSD) item fit statistic in item response models are studied. It is shown that RMSD estimates will indicate even misfit for items whose parametric assumption of the item response function is correct [...] Read more.

In this article, statistical properties of the root mean square deviation (RMSD) item fit statistic in item response models are studied. It is shown that RMSD estimates will indicate even misfit for items whose parametric assumption of the item response function is correct (i.e., fitting items) if some item response functions in the test are misspecified. Moreover, it is demonstrated that the RMSD values of misfitting and fitting items depend on the proportion of misfitting items. We propose three alternative bias-corrected RMSD estimators that reduce the bias for fitting items. However, these alternative estimators provide slightly negatively biased estimates for misfitting items compared to the originally proposed RMSD statistic. In the numerical experiments, we study the case of a misspecified one-parameter logistic item response model and the behavior of the RMSD statistic if differential item functioning occurs. Full article

(This article belongs to the Section Mathematical Sciences)

13 pages, 2675 KiB

Open AccessEditorial

Some Asian Women Pioneers of Chemistry and Pharmacy

by Juan Núñez Valdés, Fernando de Pablos Pons and Antonio Ramos Carrillo

Foundations 2022, 2(2), 475-487; https://doi.org/10.3390/foundations2020031 - 13 May 2022

Cited by 1 | Viewed by 3317

Abstract

At present, several countries on the Asian continent are still very closed off to the idea of allowing not only the work of women, but also even the fact that they can study university degrees and, after finishing them, go on to practice [...] Read more.

At present, several countries on the Asian continent are still very closed off to the idea of allowing not only the work of women, but also even the fact that they can study university degrees and, after finishing them, go on to practice their professions. In addition, if we go back to the beginning of the 20th century, this situation was even more serious. However, this was not an impediment for some women from these countries to achieve their goals of pursuing higher education and then serving society with their work. This article is dedicated to showing the biographies of three of them, the Indian chemist Asima Chatterjee and Philippine pharmacists Matilde S. Arquiza and Filomena Francisco. The most relevant features of their personal and professional lives are presented and previous biographies about them are completed. The main objective of this work is to show these figures to society and hold them up as references to other people, and the methodology followed has been the search for data about their lives and work that would allow us to complete the previous existing biographies about them. A brief biography on Janaki Ammal, the first Indian woman to obtain a doctorate, is also included. Full article

(This article belongs to the Section Chemical Sciences)

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18 pages, 333 KiB

Open AccessArticle

Location, Separation and Approximation of Solutions for Quadratic Matrix Equations

by Miguel Á. Hernández-Verón and Natalia Romero

Foundations 2022, 2(2), 457-474; https://doi.org/10.3390/foundations2020030 - 12 May 2022

Viewed by 1306

Abstract

In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This study allow [...] Read more.

In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This study allow us to determine domains of existence and uniqueness of solutions, and therefore to locate and separate the solutions. Another goal is to approximate a solution of the quadratic matrix equation. For this, we consider iterative processes of fixed point type. So, analyzing the convergence of these iterative processes of fixed point type, we locate, separate and approximate solutions of quadratic matrix equations. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

14 pages, 3702 KiB

Open AccessEditorial

Gertrude Belle Elion, Chemist and Pharmacologist, Discoverer of Highly Relevant Active Substances

by Juan Núñez Valdés, Fernando de Pablos Pons and Antonio Ramos Carrillo

Foundations 2022, 2(2), 443-456; https://doi.org/10.3390/foundations2020029 - 6 May 2022

Cited by 1 | Viewed by 2661

Abstract

Gertrude Belle Elion was a woman who had to overcome many difficulties to achieve her dream of studying to be able to cure illnesses, especially those of the heart. These difficulties were imposed both by the limited economic resources of herself and her [...] Read more.

Gertrude Belle Elion was a woman who had to overcome many difficulties to achieve her dream of studying to be able to cure illnesses, especially those of the heart. These difficulties were imposed both by the limited economic resources of herself and her family, which did not allow her to pay the academic fees of the university in which she wanted to enroll, as well as gender, since she also had to fight against inequalities of that type prevalent in the society of her time. However, and despite these obstacles, she managed to graduate in Chemistry, based on interest, effort and tenacity, and later began a research career full of successes, which led her to discover relevant active substances which allow her to be awarded the Nobel Prize in Physiology or Medicine in 1988. This article presents the most relevant features of her personal and professional life and completes previous biographies about her life. Its main objective is to reintroduce her to society and put her as a reference to other people. The methodology followed has been the search for those data about her life and work that would allow completing the previous existing biographies about her. Full article

(This article belongs to the Section Medical Sciences)

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34 pages, 433 KiB

Open AccessReview

Stochastic Interpretation of Quantum Mechanics Assuming That Vacuum Fields Are Real

by Emilio Santos

Foundations 2022, 2(2), 409-442; https://doi.org/10.3390/foundations2020028 - 4 May 2022

Cited by 3 | Viewed by 2005

Abstract

We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied [...] Read more.

We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied comparing Hilbert space and Weyl–Wigner formalisms. This allows an intuitive picture of entanglement to be obtained as a correlation between field fluctuations in distant places, involving the vacuum fields. The analysis shows that the Bell definition of local realism is not general enough, whence the reported violation of Bell inequalities does not refute local realism. Full article

(This article belongs to the Section Physical Sciences)

10 pages, 257 KiB

Open AccessArticle

Medical Records in Community Pharmacies: The Cases of UK and Australia

by Carla Pires

Foundations 2022, 2(2), 399-408; https://doi.org/10.3390/foundations2020027 - 23 Apr 2022

Cited by 3 | Viewed by 2637

Abstract

Background: Lack of access to patients’ digital health records by community pharmacists can negatively impact pharmaceutical care. Access to these records by community pharmacists is only available in some countries. Thus, the study aim was to compare and discuss the shared patients’ health [...] Read more.

Background: Lack of access to patients’ digital health records by community pharmacists can negatively impact pharmaceutical care. Access to these records by community pharmacists is only available in some countries. Thus, the study aim was to compare and discuss the shared patients’ health records between the National Health Systems and Community Pharmacies in UK and Australia. Methods: Two platforms were selected: Summary Care Records (SCR) (UK) and My Health Record (MyHR) (Australia). A qualitative and descriptive study was carried out. The type of shared health records was collected in public/official websites. Qualitative classifiers/descriptors were created to classify the shared health records. Results: The common classifiers/descriptors between both SCR and MyHR were medicines, medicines/immunization, and medical history. However, MyHR seems to comprise more details/information, such as patient’s discharge summaries, specialist letters, or documents to communicate significant patient information from one healthcare provider to another. Conclusion: Community pharmacists can update or consult SCR and MyHR to provide direct patient care in UK and Australia, respectively. The profile of shared health records with community pharmacies was not equal between SCR (UK) and MyHR (Australia). More studies are recommended to evaluate the benefits and risks of using these platforms on patients’ outcomes. Full article

22 pages, 375 KiB

Open AccessArticle

Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions

by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Foundations 2022, 2(2), 377-398; https://doi.org/10.3390/foundations2020026 - 22 Apr 2022

Cited by 4 | Viewed by 1643

Abstract

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the

ψ

-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given [...] Read more.

In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the

ψ

-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given problem is first converted into an equivalent fixed point problem, which is then solved by means of the standard fixed point theorems. The Banach contraction mapping principle is used to establish the existence of a unique solution, while the Krasnosel’skiĭ and Schaefer fixed point theorems as well as the Leray–Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multivalued analogue of the problem at hand. The existence results for convex- and nonconvex-valued multifunctions are respectively proved by means of the Leray–Schauder nonlinear alternative for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)

15 pages, 401 KiB

Open AccessArticle

Onicescu’s Informational Energy and Correlation Coefficient in Exponential Families

by Frank Nielsen

Foundations 2022, 2(2), 362-376; https://doi.org/10.3390/foundations2020025 - 19 Apr 2022

Cited by 2 | Viewed by 3998

Abstract

The informational energy of Onicescu is a positive quantity that measures the amount of uncertainty of a random variable. However, contrary to Shannon’s entropy, the informational energy is strictly convex and increases when randomness decreases. We report a closed-form formula for Onicescu’s informational [...] Read more.

The informational energy of Onicescu is a positive quantity that measures the amount of uncertainty of a random variable. However, contrary to Shannon’s entropy, the informational energy is strictly convex and increases when randomness decreases. We report a closed-form formula for Onicescu’s informational energy and its associated correlation coefficient when the probability distributions belong to an exponential family. We show how to instantiate the generic formula for several common exponential families. Finally, we discuss the characterization of valid thermodynamic process trajectories on a statistical manifold by enforcing that the entropy and the informational energy shall vary in opposite directions. Full article

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14 pages, 284 KiB

Open AccessArticle

Extending King’s Method for Finding Solutions of Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Foundations 2022, 2(2), 348-361; https://doi.org/10.3390/foundations2020024 - 18 Apr 2022

Viewed by 1414

Abstract

King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least five times differentiable functions. Consequently, these articles [...] Read more.

King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least five times differentiable functions. Consequently, these articles provide no information that can be used to solve equations involving functions that are less than five times differentiable, although King’s method may converge. That is why the new analysis uses only the operators and their first derivatives which appear in King’s method. The article contains the semi-local analysis for complex plane-valued functions not presented before. Numerical applications complement the theory. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

10 pages, 362 KiB

Open AccessArticle

Extending the Local Convergence of a Seventh Convergence Order Method without Derivatives

by Ioannis K. Argyros, Debasis Sharma, Christopher I. Argyros and Sanjaya Kumar Parhi

Foundations 2022, 2(2), 338-347; https://doi.org/10.3390/foundations2020023 - 12 Apr 2022

Viewed by 1461

Abstract

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, [...] Read more.

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, in our convergence theory, we only use the first derivative. Thus, in contrast to previously derived results, we obtain conclusions on calculable error estimates, convergence radius, and uniqueness region for the solution. As a result, we are able to broaden the utility of this efficient method. In addition, the convergence regions of this scheme for solving polynomial equations with complex coefficients are illustrated using the attraction basin approach. This study is concluded with the validation of our convergence result on application problems. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)

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Foundations, Volume 2, Issue 2 (June 2022) – 12 articles

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