Search Results (126)

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

The thin and straight horn of the ultrasonic transducer is located in the center of the thick transducer, so that the tool tip of the ultrasonic vibration turning tool holder cannot be located on the outermost side of the entire tool holder, which leads to the structural interference between the tool holder and the part during turning. In order to solve this problem, this paper proposes a longitudinal-bending elliptical vibration ultrasonic transducer with a bending horn for ultrasonic vibration-assisted cutting (UVAC). The designed transducer can be used for the partial separation continuous high-speed elliptic ultrasonic vibration cutting (HEUVC) of external surface and internal cavity. The ultrasonic vibration amplitude of the transducer can meet the needs of HEUVC. When using an ultrasonic transducer with a bending horn for HEUVC, compared with conventional cutting (CC), HEUVC can improve the tool life by about 50%. Full article

Estimating heterogeneous treatment effects plays a vital role in many statistical applications, such as precision medicine and precision marketing. In this paper, we propose a novel meta-learner, termed RXlearner for estimating the conditional average treatment effect (CATE) within the general framework of meta-algorithms. RXlearner enhances the weighting mechanism of the traditional Xlearner to improve estimation accuracy. We establish non-asymptotic error bounds for RXlearner under a continuity classification criterion, specifically assuming that the response function satisfies Hölder continuity. Moreover, we show that these bounds are achievable by selecting an appropriate base learner. The effectiveness of the proposed method is validated through extensive simulation studies and a real-world data experiment. Full article

This study explores the relationship between nutrient concentration (NC) and epidermal thickness (d) of the leaves of hydroponically grown Brassica rapa and its attenuation coefficients (m) using portable Time-Domain Optical Coherence Tomography (TD-OCT), which is a non-invasive imaging technique that uses low-coherence interferometry to generate axial scans of plants’ leaves by measuring the time delay and intensity of backscattered light. The portable TD-OCT system in this study has an axial and lateral resolution of 7 m and 3 m, respectively, a scanning depth of 12 mm, and a 1310 nm Super Luminescent Diode (SLD). Several studies suggest that the differences in d and m are related to nutritional, physiological, and anatomical status. The study used the Kratky method, a simple non-circulating hydroponic system, to cultivate Brassica rapa with varying NC (25%, 50%, 75%, 100% (control), and 125%). Each treatment group used two plants. The TD-OCT sample probe was placed on a fixed holder and was oriented vertically so that light was directed downward onto the leaf’s surface to obtain the depth profile (A-scan). The distance between the probe and the leaf was adjusted to obtain the optimum interference signal. Five averaged A-scans were obtained per leaf on the 7th, 18th, and 21st days post nutrient exposure. The logarithm of the averaged A-scan is linearly fitted to extract m. The results showed a positive correlation between NC and m, which suggests that plants produce more chlorophyll and develop denser cells and increase m. There was no correlation obtained between NC and d. The study demonstrates the potential of TD-OCT as a non-destructive tool for assessing plant health and monitoring growth dynamics in hydroponic systems and m as a sensitive indicator of plant health as compared to d. The continued exploration of TD-OCT applications in agriculture can contribute to improving crop management strategies and promoting sustainable food production practices. Full article

Vaccine marketing authorization holders (MAHs) are responsible for the conduction of global vaccine pharmacovigilance on their vaccine products. A safety signal is detected when a new adverse event (AE) or aspect of an AE occurs after exposure to the vaccine and warrants further investigation to determine whether a causal association may exist. Signal detection and evaluation (signal management) begins at the start of vaccine development, before an MAH submits an application for authorization to regulatory authorities, continues through the course of all clinical trials, and carries on beyond development into the post-marketing phase. As long as the vaccine remains authorized anywhere in the world, pharmacovigilance continues. During the time that the COVID-19 vaccine became widely available after authorization and approval, clinical trials were also ongoing, and therefore all clinical development and post-authorization safety information was closely monitored for safety by the MAH. MAH pharmacovigilance activities were adapted to manage the unprecedented volume of safety information that became available within a very short timeframe following worldwide vaccination campaigns. No vaccine had previously been administered to such a large number of individuals in such a short time, nor had there previously been a public health vaccine experience that was the subject of so many medical and non-medical writings. The MAH’s COVID-19 vaccine signal detection methods included the continuous review of accruing clinical trial data and the quantitative and qualitative analyses of spontaneously reported experiences. Review of published and unpublished medical literature and epidemiology-based analyses such as observed vs. expected analysis based on reported adverse events following immunization (AEFIs) played key roles in pharmacovigilance and signal management. All methods of signal detection and evaluation have caveats, but when considered in totality, can advance our understanding of a vaccine’s safety profile and therefore the risk–benefit considerations for vaccinating both individuals and large populations of people. All COVID-19 vaccines authorized for use were subject to an unprecedented level of pharmacovigilance by their individual MAHs, national regulatory authorities, public health organizations, and others during the years immediately following regulatory authorization and full approval. The intense worldwide focus on pharmacovigilance and the need for MAHs and regulatory/health authorities to quickly evaluate incoming safety information, spurred frequent and timely communications between national and regional health authorities and between MAHs and regulatory/health authorities, spotlighting a unique opportunity for individuals committed to patient safety to share important accruing safety information in a collegial and less traditionally formal manner than usual. The global pandemic precipitated by the SARS-CoV-2 virus created a significant impetus for MAHs to develop innovative vaccines to change the course of the COVID-19 pandemic. Pharmacovigilance also had to meet unprecedented needs. In this article, unique aspects of COVID-19 vaccine pharmacovigilance encountered by one MAH will be summarized. Full article

The temperature of the tools and the moisture content of the material play a significant role in the 3D forming of paperboard in terms of the degree of forming and the quality of the formed part. It is known that different forming mechanisms act within the paperboard in different areas of the deep drawing tools during the deep drawing of paperboard and that the success of the forming process is also based on a dynamic interaction between material moisture and tool surface temperature. However, it has not yet been investigated how the forming parameters can be influenced by an individually adjustable temperature for the individual tool areas and how they influence the complex interaction with the moisture content of the paperboard during the forming process. Due to the inhomogeneity of the natural fiber network of paperboard, rapid and directed temperature changes of the tools are also of interest in order to be able to react quickly to variations of material properties in order to prevent frequent process failure within a continuous production. In this paper, test tools with individually controllable heating zones were developed and the use of different heating technologies to improve the rate of temperature change was analyzed. These tools were used to investigate the influence of temperature in the individual sections of the deep drawing process and how the moisture content can be specifically controlled during the process. It was found that with modern heating technology, the deep-drawing tools can be tempered significantly faster and that a temperature difference between the blank holder zone and the drawing cavity zone has a positive influence on the formability and the fixation of the shape of the part produced. This effect was further enhanced by the fact that, thanks to the temperature tailored tool, it was possible to work with a very high moisture content of the paperboard. Full article

A new class of Gaussian random fields is introduced in this article, described as additive Brownian sheets (ABSs), which can be regarded as a type of generalized Brownian sheet encompassing Brownian motions, Brownian sheets, and additive Brownian motions. The existence, joint continuity and the Hölder law of the local times for ABSs are derived under certain conditions, and some results of the intersection local times for two independent Brownian sheets are also given as special cases. Furthermore, the intersection local times for two independent Brownian sheets in a Hida distribution is proved through white noise analysis, and the Wiener chaos expansion of the intersection local times is expressed in terms of S-transform. Additionally, the large deviations for the intersection local time of two independent Brownian sheets are established. The multi-parameter Gaussian random fields have become a core tool for complex system analysis due to their flexible multidimensional modeling capabilities. With the improvement of computational efficiency and interdisciplinary integration, the ABS constructed in this article will unleash greater potential in fields such as metaverse simulation, financial mathematics, climate science, precision medicine, quantum physics, and string theory. Full article

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the $\varphi$-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for $\varphi \left(\mathfrak{t}\right)$ = $\mathfrak{t}$, $\mathfrak{a}=1$. Our findings match the Riemann–Liouville fractional operator for $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=0$. They agree with the Hadamard and Caputo–Hadamard fractional operators when $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=0$ and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=1$, respectively. Second, regarding the space, we are make generalizations for the case $\mathfrak{p}=2$. Full article

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems. Full article

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: ${\displaystyle \sum _{\left|\alpha \right|=1,2}}{(-1)}^{\left|\alpha \right|}{\mathrm{D}}^{\alpha }\left[A_{\alpha }(x,u,{\mathrm{D}}^{1}u,{\mathrm{D}}^{2}u)-E_{\alpha }\left(x\right) {\left|u\right|}^{\lambda (p_{\alpha }-1)} signu\right]=f$ in $\Omega$, where the coefficients $A_{\alpha }$ satisfy a strengthened degenerate coercivity condition. Full article

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent. Full article

The Secretary of Public Health (SSP) faces a looming skills gap due to retirements and rotations of civil service staff. Critical knowledge retention is crucial across all generational cohorts due to the retirement and turnover of workers. This study develops a protocol that addresses the knowledge retention needs of the four generations (Baby Boomers, X, Y, Z) that coexist in the workforce to ensure the continuity of the Public Health Secretariat. The objective of the study is to develop a protocol for the management, transfer, and retention of critical knowledge. A scoping review is conducted in Scopus and Web of Science to develop the protocol, to identify critical knowledge workers through tool scores. The instrument developed in this research includes two pilots on Baby Boomer and Millennial workers. Both workers had critical and essential knowledge for the continuity of the organisation. The Baby Boomer worker presented a higher amount of tacit, operational, and individually owned knowledge, while the Millennial worker showed a predominance of tacit technological knowledge. This protocol provides a practical and adaptable approach to identifying and prioritising critical knowledge holders, allowing organisations to map and determine the amount of essential knowledge within the workforce. An important limitation of the study is the small sample of workers who participated in the pilot test of the protocol. Further research is therefore recommended in other public administrations and across all generations in employment. Full article

We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called ${\mathsf{\Phi }}_s$ functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ${\mathsf{\Phi }}_M$ functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications. Full article

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions $C[a,b]$ equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space. Full article

Following the work on non-stationary fractal interpolation (Mathematics 7, 666 (2019)), we study non-stationary or statistically self-similar fractal interpolation on the Sierpiński gasket (SG). This article provides an upper bound of box dimension of the proposed interpolants in certain spaces under suitable assumption on the corresponding Iterated Function System. Along the way, we also prove that the proposed non-stationary fractal interpolation functions have finite energy. Full article