This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the $p$-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the "broom topology'' type.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

We study the deterministic dynamics of N point particles moving at constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback-control on the particles, inverting the horizontal component of their velocities, when their number in the channel exceeds a fixed threshold. Such a bounce--back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed both for rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted.

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.

Sensory integration is an essential function of human life, and its decline causes difficulties in the quality of life, especially in older adults. In this study, we proposed an arm-reaching task based on a virtual reality head-mounted display (VR-HMD) system to assess sensory integration in daily life, and examined whether reaching task performance was associated with age and resting-state functional connectivity (rsFC) between the brain regions of interest involved in sensory integration. We hypothesized that performance during a reaching task with multiple cognitive loadings would be affected by aging. The gap angle between the reaching hand and target positions was compared between older and middle-aged adults as a measure of sensory integration function. In addition, rsFC, which was significantly correlated with the gap angles of individual participants in the older adult group, was identified using resting-state functional magnetic resonance imaging data. The mean gap angle was significantly larger in the older adults than in the middle-aged group and was also significantly correlated with connections between the left primary motor area and the left inferior temporal gyrus, temporo-occipital part. Because the connectivity between the areas is known to be related to visuomotor integration, the results suggest the involvement of visuomotor integration in the decline of sensory integration function in old age and the validity of the gap angle during the proposed VR-HMD task as an index for the decline in function.

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.

The paper deals with exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k that include base-two exponential functions of k have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Autonomous vehicle technology, employing image sensors for road marking recognition, is largely dependent on the Lane Departure Warning System (LDWS) to prevent vehicular deviations from driving lanes. The effectiveness of image-based lane recognition, however, is susceptible to environmental influences like weather conditions and the surrounding road environment. This study investigates the impact of road marking retro-reflectivity on LDWS under varied environmental conditions. Conducted at the Yeoncheon SOC Demonstration Research Center, the experiments encompassed a spectrum of weather scenarios, incorporating rainfall and shifts between day and night lighting. Controlled wear was applied to white, yellow, and blue road markings, followed by measuring their retro-reflectivity at different stages of degradation. The LDWS's performance was evaluated by assessing the recognition rates of these markings under these diverse environmental settings. The study reveals that enhanced retro-reflectivity significantly boosts LDWS detection, especially in challenging weather conditions. Furthermore, the research led to the development of a simulation framework to analyze the cost-effectiveness of road marking maintenance, aligning it with the safety needs of autonomous vehicles. This framework suggests the necessity for updated road marking guidelines that cater to the advanced needs of driver assistance systems and autonomous vehicle technology.

Collaborative action between families and health professionals, in this case nurses, doctors, nutritionists, pharmacies in forming harmonious support is the philosophy of family centered nursing care which aims to involve families as the main focus in care. The aim of this literature review is to identify family-centered family member care: a literatu review. This language method uses literature reviews which are summaries of 10 articles in the publication years of 2020-2021 on search 4 databased electronic searches containing namely Scopus, ProQuest, Pubmed, and Scient Direct. This review used prisms. The eligibility of these studies were from its title, abstract, research methodology, results and discussion. The results of the review were presented in narrative form. The results of a review of 10 articles found that the form family centered care Conclusion: The family is considered a partner in the care of other family members. The concept of family centered care is a philosophy in nursing where the role of the family is very important in caring for family members who are sick.

In this paper, we present a general version of polygonal fitting problem called Unconstrained Polygonal Fitting (UPF). Our goal is to represent a given 2D shape S with an N-vertex polygonal curve P with a known number of vertices, so that the Intersection over Union (IoU) metric between S and P is maximized without any assumption or prior knowledge of the object structure and the location of the N-vertices of P that can be placed anywhere in the 2D space. The search space of the UPF problem is a superset of the classical polygonal approximation (PA) problem, where the vertices are constrained to belong in the boundary of the given 2D shape. Therefore, the resulting solutions of the UPF may better approximate the given curve than the solutions of the PA problem. For a given number of vertices N, a Particle Swarm Optimization (PSO) method is used to maximize the IoU metric, which yields almost optimal solutions. Furthermore, the proposed method has also been implemented under the equal area principle so that the total area covered by P is equal to the area of the original 2D shape to measure how this constraint affects IoU metric. The quantitative results obtained on more than 2,800 2D shapes included in two standard datasets quantify the performance of the proposed methods and illustrate that their solutions outperform baselines from the literature.

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

This paper addresses the development of knowledge and assessment-centered learning approaches within a reflective learning framework in a first year physics class in a university faculty. The quality of students’ reflections was scored using a Self-reporting Reflective Learning Appraisal Questionnaire at the end of each learning approach. The results showed the differences between the approaches based on reflections on the learning control through self-knowledge, by connecting experience and knowledge, as well as through self-reflection and self-regulation. Assessment-centered activities fundamentally help students identify aspects of their attitudes towards, as well as regulate, their sustainability learning education.

Vehicle identification is an important task in traffic monitoring because it allows for efficient inference and provides a cause for action. Vehicle classification via deep learning and other approaches such as segmentation is a critical tool for re-identification. In this paper, instance segmentation is used to identify vehicle makes with license plate detection, allowing for better unique vehicle recognition for re-identification. A dataset is annotated and modified, for example, by segmenting it with polygonal bounding boxes that capture the vehicle's unique frontal features. In addition, license plate localization is performed. The results showed improved classification as well as a high mAP for the dataset when compared to previous approaches based on CNN and deformed CNN. Furthermore, a deep residual network and fully connected layer-based classification were utilized as the backbone for feature representation. Instance segmentation detects objects by segmenting and classifying regions of interest. The imbalance in the dataset is resolved using a mosaic-tiled approach, which produces greater precision than other approaches.

The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

Patients with a psychotic disorder frequently report oral health problems, while mental health nurses (MHNs) seem not to be fully aware of these problems and the risk factors. Therefore, this study aims to develop supportive tools for MHNs regarding oral health to increase sensitization among MHNs and to provide MHNs with knowledge to recognize (potential) oral health problems in patients with a psychotic disorder. We used a human-centered design in which the user, MHNs as well as experts by experience, were placed in the centre of the research process. Problems and needs in MHNs working with patients with a psychotic disorder were addressed. To identify key issues of problem as well as the needs in terms of resources, we started with focus groups (n = 9). We analysed the data thematically based on context of patients and MHNs in practice regarding oral health, preferred oral health tools focused on MHN, and intended outcomes of tools. A Multi Criteria Decision Matrix was developed and analysed (n = 9) to identify the most optimal and viable solution based on established criteria and issues that are prevalent in the work of MHNs. Most promising result was the development of a brochure with an awareness screener. The brochure with awareness screener was developed as a low-fidelity prototype for MHNs regarding oral health in patients with a psychotic disorder, based on the latest scientific evidence. After testing it, the feasibility was tested through semi-structured interviews (n=19). MHNs and experts by experience were satisfied with the tool and provided recommendations for adjustments to the content. Significant augmentations to the brochure include a clinical lesson and a toothbrush with toothpaste for patients. We can conclude that a brochure with an awareness screener is feasible for MHNs. Future steps aiming to further refine and optimize care processes for oral health tools in MHNs, should take refining eligibility criteria for psychiatric populations and the language level of the target group of a tool into account.

The pursuit of sustainable wellbeing is one of the research objectives of positive experience design. Driven by this goal, the purpose of this paper is to provide an impact-centered sustainable positive experience design model. Firstly, the literature review method was used to define the research status and concept of impact-centered sustainable design. Secondly, an impact-centered sustainable positive experience design model was constructed, and relevant formulas for concept generation and concept evaluation were proposed. Thirdly, design verification was conducted through a workshop. Finally, the Technology Acceptance Model (TAM) questionnaire was used to evaluate and discuss the design model. An impact-centered sustainable positive design model was proposed, which includes the important impact dimensions of sensory experience and meaningful experience on users’ quality of life at different levels: healthy living (pleasure index, health behavior), harmonious community (social connectivity, social contribution), and livable environment (living environment, environmental contribution). Based on positive experience related theory, this study takes long-term impacts as the starting point for sustainable positive experience design, which helps designers to generate design concepts from a systematic and long-term perspective.

Based on a measuring system to determine the statistical heterogeneity of individual polygons we propose a method to use polygonal shape patterns as a source of data in order to determine the Shannon entropy of biological organizations. In this research, the term entropy is a particular amount of data related with levels of spatial heterogeneity in a series of different geometrical meshes and sets of random polygons. We propose that this notion of entropy is important to measure levels of information in units of bits, measuring quantities of heterogeneity in geometrical systems. In fact, one important result is that binarization of heterogeneity frequencies yields a supported metric to determine geometrical information from complex configurations. Thirty-five geometric aggregates are tested; biological and non-biological, in order to obtain experimental results of their spatial heterogeneity which is verified with the Shannon entropy parameter defining low particular levels of geometrical information in biological samples. Geometrical aggregates (meshes) include a spectrum of organizations ranging from cell meshes to ecological patterns. Experimental results show that a particular range (0.08 and 0.27) of information is intrinsically associated with low rates of heterogeneity. We conclude it as an intrinsic feature of geometrical organizations in multi-scaling biological systems.

Introduction: Family-centered care (FCC) is currently taking a greater role in health care, due to the increasing empowerment of the parental experience. This literature aims to find out the study of the philosophy of child nursing services based on Family Centered CareMethode: The study of this literature through the search of scientific publications ranges from 201 7-2022. The databases used are Pubmed, Science Direct, and Proquest. The literature search was conducted according to the topic with four keyword groups based on Medical Subject Heading (MeSH) and combined with Boolean operators AND, OR and NOT, keywords Child AND Family AND Centered AND Care. Result: The study of the philosophy of child nursing services based on Family Centered Care is conducted based on the study of ontology, epistomotogi and axiology. Family-centered care (FCC) is a philosophy that emphasizes partnerships between parents and health care staff. The FCC's basic principles in neonatal intensive care units (NICs) are unlimited parental presence and parental participation, shared responsibility and decision-making about infant hospital care, and open communication between parents and staff. A family-centered treatment approach has benefits such as maintaining a bond between the child and the family, allowing the family to engage in child care, and minimizing the negative effects of hospitalization on the child and family.Conclusion: FCC is beneficial for nurses in terms of efficiency and effectiveness of time in the care process and increases professional satisfaction. Therefore, the implementation of the FCC is more effective and the satisfaction of clients and families is more increased.

In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between modified p-Leonardo, p-Leonardo-Lucas, p-Leonardo numbers and Fibonacci, Lucas numbers.

In this paper, we consider the fourth-order parabolic equations with integer and fractional order time derivatives with Neumann boundary conditions. The integer order time derivatives are approximated by backward Euler difference quotients, and the fractional order time derivatives are approximated by L1 interpolation. We propose the block-centered finite difference scheme for fourth-order parabolic equations of integer and fractional order time derivatives. We prove the stability of the block-centered finite difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are given to verify the effectiveness of the block-centered finite difference scheme.

Schizophrenia continues to be an illness with poor outcome. Most mechanistic changes occur many years before the first episode of schizophrenia; these are not reversible after the illness onset. A developmental mechanism that is still modifiable in adult life may center on intracortical glutathione (GSH). A large body of pre-clinical data has suggested the possibility of notable GSH-deficit in a subgroup of patients with schizophrenia. Nevertheless, studies of intracortical GSH are not conclusive in this regard. In this review, we highlight the recent ultra-high field magnetic resonance spectroscopic studies linking GSH to critical outcome measures across various stages of schizophrenia. We discuss the methodological steps required to conclusively establish or refute the persistence of GSH-deficit subtype and clarify the role of the central antioxidant system in disrupting the brain structure and connectivity in the early stages of schizophrenia. We propose in-vivo GSH quantification for patient selection in forthcoming antioxidant trials in psychosis. This review offers directions for a promising non-dopaminergic early intervention approach in schizophrenia.

This paper presents a steady-state and transient heat conduction analysis framework using the polygonal scaled boundary finite element method (PSBFEM) with polygon/quadtree meshes. The PSBFEM is implemented with commercial finite element code Abaqus by the User Element Subroutine (UEL) feature. The detailed implementation of the framework, defining the UEL element, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from heat conduction are solved to validate the proposed implementation. Results show that the PSBFEM is reliable and accurate for solving heat conduction problems. Not only can the proposed implementation help engineering practitioners analyze the heat conduction problem using polygonal mesh in Abaqus, but it also provides a reference for developing the UEL to solve other problems using the scaled boundary finite element method.

In this paper, we investigate the generalized Padovan sequences and we deal with, in detail, four special cases, namely, Padovan, Perrin, Padovan-Perrin and modified Padovan sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Number is based in special edges, is introduced. The kind of natural extension from edges toward vertex and the set of vertices in the way that, the final notion is number, is studied. The result is obtained which is about the study on the classes of graphs in the matter of new notions. There is the extended notion about having edge amid two vertices toward having some edges in the word of neighbor and in another stage going into the atmosphere of having consecutive edges in the terminology of path and in the upper vision going on the notion about path with jargon and buzzword of distance as if the minimal vertices has concluded the new notions with the word, number.

New setting is introduced to study “closing numbers” and “super-closing numbers” as optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. In this way, some approaches are applied to get some sets from (Neutrosophic)n-SuperHyperGraph and after that, some ideas are applied to get different types of super-closing numbers which are called by optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. The notion of dual is another new idea which is covered by these notions and results. In the setting of dual, the set of super-vertices is exchanged with the set of super-edges. Thus these results and definitions hold in the setting of dual. Setting of neutrosophic n-SuperHyperGraph is used to get some examples and solutions for two applications which are proposed. Both setting of SuperHyperGraph and neutrosophic n-SuperHyperGraph are simultaneously studied but the results are about the setting of n-SuperHyperGraphs. Setting of neutrosophic n-SuperHyperGraph get some examples where neutrosophic hypergraphs as special case of neutrosophic n-SuperHyperGraph are used. The clarifications use neutrosophic n-SuperHyperGraph and theoretical study is to use n-SuperHyperGraph but these results are also applicable into neutrosophic n-SuperHyperGraph. Special usage from different attributes of neutrosophic n-SuperHyperGraph are appropriate to have open ways to pursue this study. Different types of procedures including optimal-super-set, and optimal-super-number alongside study on the family of (neutrosophic)n-SuperHyperGraph are proposed in this way, some results are obtained. General classes of (neutrosophic)n-SuperHyperGraph are used to obtains these closing numbers and super-closing numbers and the representatives of the optimal-super-coloring sets, optimal-super-dominating sets and optimal-super-resolving sets. Using colors to assign to the super-vertices of n-SuperHyperGraph and characterizing optimal-super-resolving sets and optimal-super-dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on n-SuperHyperGraph to get new results about closing numbers and super-closing numbers alongside sets in the way that some closing numbers super-closing numbers get understandable perspective. Family of n-SuperHyperGraph are studied to investigate about the notions, super-resolving and super-coloring alongside super-dominating in n-SuperHyperGraph. In this way, sets of representatives of optimal-super-colors, optimal-super-resolving sets and optimal-super-dominating sets have key role. Optimal-super sets and optimal-super numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal-super ones. Simultaneously, three notions are applied into (neutrosophic)n-SuperHyperGraph to get sensible results about their structures. Basic familiarities with n-SuperHyperGraph theory and neutrosophic n-SuperHyperGraph theory are proposed for this article.

In the paper, by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.

According to the International Agency for Research on Cancer (IARC) of the World Health Organization (data of year 2020), Breast Cancer (BC) is one of the most common cancer types worldwide, with large geographical variations occurring between countries and world regions and highest incidence rates in countries that have undergone economic transition. The risk factors for BC include women ageing, genetic mutations, reproductive history, dense breast tissues, personal history of BC or specific non-cancerous breast diseases, family history of breast or ovarian cancer, previous treatment using radiation therapy, and exposure to hormone-like drugs such as diethylstilbestrol (DES). Additional risk factors include being overweight or having obesity after menopause, and taking hormones. MicroRNAs (miRNAs or miRs) are small non-coding RNA molecules that have an essential role in gene expression and are significantly linked to the tumor development and angiogenesis process in different types of cancer, including BC. In the last years, the knowledge about miRNAs role in BC has significantly improved, and complex interactions among coding and non-coding RNA has been elucidated. In this context, an increasing number of papers had been published regarding the role of miR-125 in BC. In this review, we summarize the state-of-the-art about this research topic in addition to elaborating on the need to set novel ethical and legal standards for the governance of such innovations in healthcare.

Brain-computer interfaces (BCI) from electroencephalography (EEG) provide a practical approach to support human-technology interaction. In particular, motor imagery (MI) is a widely-used BCI paradigm that guides the mental trial of motor tasks without physical movement. Here, we present a deep learning methodology, named Kernel-based Regularized EEGNet (KREEGNet), leveled on Centered Kernel Alignment and Gaussian Functional Connectivity, explicitly designed for EEG-based MI classification. The approach proactively tackles the challenge of intra-subject variability brought on by noisy EEG records and the lack of spatial interpretability within end-to-end frameworks applied for MI classification. KREEGNet is a refinement of the widely accepted EEGNet architecture, featuring an additional kernel-based layer for regularized Gaussian functional connectivity estimation based on CKA. The superiority of KREEGNet is evidenced by our experimental results from binary and multi-class MI classification databases, outperforming the baseline EEGNet and other state-of-the-art methods. Further exploration of our model’s interpretability is conducted at individual and group levels, utilizing classification performance measures and pruned functional connectivities. Our approach is a suitable alternative for interpretable end-to-end EEG-BCI based on deep learning.

Although child-centred care is increasingly referred to within the nursing literature, a clear definition of child-centred care and clarity around the concept is yet to be achieved. The objectives of this review were to examine: 1) What constitutes the concept of child-centred care in healthcare? 2) How has the concept of child-centred care developed? 3) What is the applicability of child-centred care and what are its limitations? and 4) How does the concept of child-centred care benefit and inform children’s healthcare? In total, 2984 papers were imported for screening, following removal of duplicates and screening, 21 papers were included in the scoping review.The findings suggest that child-centred care is an emerging, ambiguous, poorly defined concept; no clear consensus exists about what constitutes child-centred care. Although it seems antithetical to argue against child-centred care, little robust evidence was identified that demonstrates the impact of, and benefit from, child-centred care. If child-centred care is to be a sustainable, convincing model to guide practice and compete with other models of care it needs to establish robust evidence of its effectiveness, impact on children and their families as well as the wider impacts on the healthcare system.

Pancreatic ductal adenocarcinoma (PDAC) a highly malignant neoplasm, is classified as one of the most severe and devastating types of cancer. PDAC is a notable malignancy that exhibits a discouraging prognosis and a rising occurrence. The interplay between diabetes and pancreatic cancer exhibits a reciprocal causation. The identified metabolic disorder has been observed to possess noteworthy consequences on health outcomes, resulting in elevated rates of morbidity. The principal mechanisms involve the suppression of the immune system, the activation of pancreatic stellate cells (PSCs), and the onset of systemic metabolic disease caused by dysfunction of the islets. From this point forward, it is important to recognize that pancreatic cancer-related diabetes (PCRD) has the ability to increase the likelihood of developing pancreatic cancer. This highlights the complex relationship that exists between these two physiological states. Therefore, we investigate into the complex domain of PSCs, elucidating their intricate signaling pathways and the profound influence of chemokines on their behavior and final outcome. In order to surmount the obstacle of drug resistance and eliminate PDAC, researchers have undertaken extensive efforts to explore and cultivate novel natural compounds of the next generation. Additional investigation is necessary in order to comprehensively comprehend the effect of PCRD-mediated apoptosis on the progression and onset of PDAC through the utilization of natural compound. This study aims to examine the potential anticancer properties of natural compound in individuals with diabetes who are undergoing chemotherapy, targeted therapy, or immunotherapy. It is anticipated that these compounds will exhibit increased potency and possess enhanced pharmacological benefits. According to our research findings, it is indicated that naturally derived chemical compounds hold potential in the development of PDAC therapies that are both safe and efficacious.

Objective: To map the literature pertaining to adult Person-Centered Rehabilitation (PCR).Data Sources: Following our previously published scoping review protocol, data were identified through: three major databases, snowball searches and expert consultation.Study Selection: Two independent reviewers identified English-language papers which addressed adult physical rehabilitation and met one or more of the six pre-defined inclusion categories for PCR content. Data Extraction: Two independent reviewers extracted key characteristics of included papers (e.g. aims, methods, participants). Quantitative methods (descriptive statistics, regression analysis) and qualitative content analyses were used to synthesize the results. Data synthesis: Of 5084 unique records initially screened, 145 papers were included: 111 empirical, including 11 systematic reviews. Empirical papers had data from 13498 clients and 3849 providers, in total. Yearly publications grew exponentially from 2000 to 2017 (r²= 0.65; p<0.01). Publications were unevenly distributed by countries (e.g. United States’ publications per population size was 45 times lower than New Zealand’s). Most papers focused in more than one profession, setting-type or health conditions, respectively 57.2%, 66.2%, and 60.7%. Finally, more than half of the empirical papers (53.2%) studied implementation of PCR approaches, including its effect. Conclusion: This scoping review synthesizes key characteristics and publication trends in the PCR literature, which is mainly empirical and growing exponentially over time. Stakeholders can use the identified trends, gaps, and literature map to guide further PCR research, and both clinical and organizational practices. The high prevalence of papers focused on multiple professions, settings-type, or health conditions reinforces the need for developing a transdisciplinary, trans-service model of PCR, which will be derived from a thematic analysis of this body of literature.

This study investigated the breakdown trend in an automated production with an aim to recommend the application of Reliability-centered maintenance (RCM) for an improved productivity via a new preventive maintenance (PM) program. Individual Section-forming machine (ISM): a glass blowing machine for making glass bottles was used as the case study of an automated production system. The machine parts and the working mechanisms were analysed with a special focus on methods of processes and procedures that will enable the ISM maintenance department run more effective and achieve its essential goal of ensuring effective machine operation and reducing machine downtime. In this work, information is provided on the steps and procedures to identify critical components of the ISM using Failure Modes and Effect Analysis (FMEA) as a tool to develop an optimal maintenance program based on the reliability data of the equipment’s functional components. A relationship was established between the failure rate of the machine components and the maintenance costs such that using the recommended PM program, an evidence of improvement in the machine’s availability, safety and cost-effectiveness will result into an increase in the company’s profit margin.

The article shows materials to the question about algebraic features of the genetic code and about the dictatorial influence of the DNA and RNA molecules on the whole organism. Presented results testify in favor that the genetic code is an algebraic code related with a wide class of algebraic codes, which are a basis of noise-immune coding of information in communication technologies. Structural features of the genetic systems are associated with hypercomplex double (or hyperbolic) numbers and with bisymmetric doubly stochastic matrices. The received results confirm that represented matrix approaches are effective for modeling genetic phenomena and revealing the interconnections of structures of biological bodies at various levels of their organization. This allows one to think that living organisms are algebraically encoded entities where structures of genetic molecules have the dictatorial influence on inherited structures of the whole organism. New described algebraic approaches and results are discussed.

In this paper, we define Tetranacci and Tetranacci-Lucas matrix sequences and investigate their properties.

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and Pentanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numbers and as special cases, we investigate Gaussian Pentanacci and Gaussian Pentanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Pentanacci numbers and Gaussian generalized Pentanacci numbers.

In this paper, we introduce the partially degenerate Changhee-Genocchi polynomials and numbers and investigated some identities of these polynomials. Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear dierential equations.

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

When presenting special relativity, it is customary to single out the so-called paradoxes. One of these paradoxes is the formal occurrence of speeds exceeding the speed of light. An essential part of such paradoxes arises from the incompleteness of the relativistic calculus of velocities. In special relativity, the additive group is used for velocities. However, the use of only group operations imposes artificial restrictions on possible computations. Naive expansion to vector space is usually done by using non-relativistic operations. We propose to consider arithmetic operations in the special theory of relativity in the framework of the Cayley–Klein model for projective spaces. We show that such paradoxes do not arise in the framework of the proposed relativistic extension of algebraic operations.

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1+1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1+2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1+2 dimensions.

The importance and near ubiquity of the Golden Ratio in disciplines like chemistry and biology is well known, but only recently has it come to light in areas pertaining to the Quantum domain. By using a modified tool-kit of hyper-complex numbers (known as Dimensional Gate Operators) and numerical analysis, we uncover a connection between hyper-dimensional objects, the Fibonacci sequence and that of quasiparticles and excitations. The results show that dimensionality increases in step with the Fibonacci sequence.

This study measured secondary students’ digital literacy using a digital game-based assessment system that was designed and developed based on the Evidence-Centered Game Design (ECGD) approach. A total of 188 secondary students constituted the valid samples in this study. Fine-grained behavioral data generated from students’ gameplay processes were collected and recorded with the assessment system. The Delphi method was used to extract feature variables related to digital literacy from the process data, and the Analytic Hierarchy Process (AHP) method was used to construct the measurement model. The assessment results of the ECGD-based assessment had a high correlation with standardized test scores, which have been shown to be reliable and valid in prior large-scale assessment studies.

Activity of the Nox1 centered NADPH oxidase complex depends on the activator NoxA1 and the organizer NoxO1. NoxO1 itself is not involved in electron transfer for superoxide formation. Instead, by creating proximity of all components in the right place NoxO1 enables a constitutive formation of reactive oxygen species by that complex. All subunits should form the complex in a 1:1:1 ratio. When analyzing different cell lines in this study, we found an unequal expression of the component on mRNA level with an excess of NoxO1. Even with plasmid-based overexpression of individual components of Nox1 centered NADPH oxidase complex results in different expression of their mRNA, with NoxO1 mRNA being best expressed. Despite an unchanged high level of NoxO1 mRNA over a wide range of transfected plasmid, protein expression is increased with accelerating plasmid concentrations. We thought to analyze the ability of NoxO1 to induce ROS formation, when present in different ratio to Nox1 and NoxA1. To this end, we used Hek293 cells with constitutive expression of Nox1 and NoxA1 transfected with increasing concentrations of NoxO1. Our results suggest that ROS formation by the Nox1 centered NADPH oxidase strongly depends on the level of NoxO1 and a surplus of NoxO1 further increases the activity of the complex.

Cyber-Physical Systems form the new backbone of digital ecosystems. Their design can be coupled with engineering activities to facilitate dynamic adaptation and (re-)configuration. Behavior-oriented technologies enable highly distributed and while coupled operation of systems. Utilizing them for digital twins as self-contained design entities with federation capabilities makes them promising candidates to develop and run highly functional CPS. In this paper we discuss mapping CPS components to behavior-based digital twin constituents mirroring integration and implementation through subject-oriented models. These models, inspired by agent-oriented system thinking can be executed and increase transparency at design and runtime. Patterns recognizing environmental factors and operation details facilitate configuration of CPS. Subject-oriented runtime support enable dynamic adaptation and federated use.

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p · n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p · n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 · n · p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 · n · p + p formula we put forth with another known formula.

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

A new generalized definition of Mersenne numbers is proposed of the form (a^n - (a-1)^n), called Global Generalized Mersenne numbers, or simply Generalized Mersenne numbers and noted GM_{a,n} where a is the base and n is the exponent, both being positive integers. The properties are investigated for prime exponents n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. In particular, it is found that, for any base a, Generalized Mersenne numbers are in general such that (GM_{a,n} - 1) are even and divisible by n, a and (a-1) for any odd prime exponent n and by (a(a-1)+1) for any prime exponent n > 5. The remaining factor is a function of triangular numbers of (a-1), specific to each prime exponent n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, allowing to show first, that (GM_{a,n} - 1) are divisible by 6, and more precisely GM_{a,n} are congruent to 1(mod 12) or 7(mod 12) depending on the congruence of the base a(mod 4); second, that (GM_{a,n} - 1) are divisible by 10 if n == 1(mod 4) and, if n == 3(mod 4), GM_{a,n} == 1(mod 10), or 7(mod 10) or 9(mod 10) depending on the congruence of the base a(mod 5); third, that all factors c_{i} of GM_{a,n} are of the form (2nf_{i}+1) with f_{i} natural integers such that c_{i} is prime itself or the product of primes of the form (2nj+1) with j natural integer; fourth, that for odd prime exponents n, all GM_{a,n} are periodically congruent to either +/-1\(mod 8) or +/-3(mod 8) depending on the congruence of the base a(mod 8); and fifth, that the factors of a composite GM_{a,n} is of the form (2nf_{i}+1) with f_{i} == u(mod 4) and u being either 0, 1, 2 or 3 depending on the congruence of the exponent n(mod 4) and on the congruence of the base a(mod 8). The potential use of Generalized Mersenne primes in cryptography is shortly addressed.

Cognitive linguistics is founded on the assumption that language is a reflection of universal cognitive principles, implying that all cognitive processes share a common origin. This assumption is known as the 'Cognitive Commitment.' The present article examines the sensory schema hypothesis, which traces all forms of expression back to sensations. This theoretical framework integrates the physicality of the body with sensations that underlie human experiences and their linguistic and non-linguistic expressions. By challenging the traditional dichotomy between mind and body, it provides a more integrated understanding.

We consider an $N$ fermion system at low temperature $T$ in which we encounter special particle number values $N_m$ exhibiting special traits. These values arise in focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling constant of the Hamiltonian, the DMs stay constant for all $N$-values, but experience sudden jumps at the $N_m$. For a quantum state described by the matrix $\rho$, its purity is expressed by $Tr \rho^2$ and then the degree of mixture is given by $1 - Tr \rho^2$, a quantity that coincides with the entropy $S_q$ for $q=2$. Thus, Tsallis entropy of index two faithfully represents the degree of mixing of a state, that is, it measures the extent to which the state departs from maximal purity . Macroscopic manifestations of the degree of mixing can be observed through various physical quantities. Our present study is closely related to properties of many-fermion systemsn that are usually manipulated at zero temperature. Here we wish to study the subject at finite temperature. Gibbs' ensemble is appealed to. Some interesting insights are thereby gained.

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

In this paper, we introduce a new class of degenerate Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of degenerate Hermite-Fubini numbers and polynomials by using generating functions.

The main purpose of this paper is to introduce a new class of $q$-Hermite-Fubini numbers and polynomials by combining the $q$-Hermite polynomials and $q$-Fubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. Also, we establish some relationships for $q$-Hermite-Fubini polynomials associated with $q$-Bernoulli polynomials, $q$-Euler polynomials and $q$-Genocchi polynomials and $q$-Stirling numbers of the second kind.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π, and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication.

In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].

In the evolving landscape of higher education, St. Michael's College (SMC) recognizes the significant impact of the COVID-19 pandemic on student welfare. This paper emphasizes the critical importance of addressing mental health, time management, and stress in the college environment, underlining their influence on academic success and personal growth. The call to action urges students to prioritize their well-being. Diverse cultural and religious perspectives, including Islam, emphasize the importance of a balanced mind and body. The wide-ranging literature review uncovers empirically supported strategies for mental health, time management, and stress reduction. Personal autonomy emerges as crucial in addressing these challenges. This study underscores the importance of these dimensions, calls for student action, and highlights the diversity of perspectives contributing to holistic well-being.

The article is devoted to applications of 2-dimensional hyperbolic numbers and their algebraic 2ⁿ-dimensional extensions in modeling some genetic and cultural phenomena. Mathematical properties of hyperbolic numbers and their bisymmetric matrix representations are described in a connection with their application to analyze the following structures: alphabets of DNA nucleobases; inherited phyllotaxis phenomena; Punnett squares in Mendelian genetics; the psychophysical Weber-Fechner law; long literary Russian texts (in their special binary representations). New methods of algebraic analysis of the harmony of musical works are proposed, taking into account the innate predisposition of people to music. The hypothesis is put forward that sets of eigenvectors of matrix representations of basis units of 2ⁿ-dimensional hyperbolic numbers play an important role in transmitting biological information. A general hyperbolic rule regarding the oligomer cooperative organization of different genomes is described jointly with its quantum-information model. Besides, the hypothesis about some analog of the Weber-Fechner law for sequences of spikes in single nerve fibers is formulated. The proposed algebraic approach is connected with the theme of the grammar of biology and applications of bisymmetric doubly stochastic matrices. Applications of hyperbolic numbers reveal hidden interrelations between structures of different biological and physical phenomena. They lead to new approaches in mathematical modeling genetic phenomena and innate biological structures.

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

The endometrium, the inner mucosal lining of the uterus, undergoes complex molecular and cellular changes across the menstrual cycle in preparation for embryo implantation. Transcrip-tome-wide analyses have mainly been utilized to study the endometrial receptivity, the prerequisite for successful implantation, with most of the studies, so far, comparing the endometrial transcriptomes between (i) secretory versus proliferative endometrium or (ii) mid-secretory versus early-secretory endometrium. In the current study, we provide the complete transcriptome description of the endometrium across the entire menstrual cycle and, for the first time, comprehensively characterize the proliferative phase of the endometrium. Our temporal transcriptome analysis includes five time points: mid-proliferative, late-proliferative (peri-ovulatory phase), early-secretory, mid-secretory, and late-secretory phases. Thus, we exhaustively unveil the transitions between the consecutive proliferative and secretory phases, highlighting their unique gene expression profiles and possible distinct biological functions. Transcriptome analysis reveals many differentially expressed genes (DEGs) across the menstrual cycle, most of which are phase-specific. As an example of the coordinated gene activity, the expression profile of his-tone-encoding genes within the HIST cluster on chromosome 6 showed an increase in cluster activity during the late-proliferative and a decline during the mid-secretory phase. Moreover, numerous DEGs are shared among all phases. In conclusion, in the current study, we delineate the endometrial proliferative phase-centered view of transcriptome dynamics across the menstrual cycle. Data analysis highlights significant transcriptomic and functional changes occurring during the late-proliferative phase – an essential transition point from the proliferative to the secretory phase. Future studies should explore how the biology of the late-proliferative phase endometrium impacts the achievement of mid-secretory endometrial receptivity or contributes to molecular aberrations leading to embryo implantation failure.

The existence of palindrome numbers of arbitrary length greater than 2 that are divisible by the prime number 37 is proven in this paper. The method to construct such a palindrome is also described. It turns out that a palindrome number divisible by 37 is also divisible by 3 and, consequently, divisible by 111.

Recently, a paper was written to reformulate and strengthen the theory of semi-structured complex numbers H (a new number set invented to enable division by zero). Whilst the paper had profound results, the application of this number set in many areas of engineering and science has not been fully explored. Consequently, the aim of this paper was to establish the number set H as a useful mathematical tool by providing an amalgam of results arising from its application in several areas of engineering and science. In the process, this paper makes four major contributions: these are (1) determining the product of the gradients of a horizontal and vertical line; (2) developing new Euler formulas relating exponential, trigonometric and hyperbolic functions in the 3D semi-structured complex number Euclidean space; (3) establishing semi-structured complex numbers as a better extension of the complex number set than quaternions; (4) resolving singularities that may arise in engineering and science equations (because of division by zero) to develop reasonable conclusions in the absence of experimental data. These results and their applications provide a firm foundation to advance the number set H as a useful mathematical tool.

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

Recently, Kim-Kim [13] have introduced polyexponential functions as an inverse to the polylogarithm functions, and constructed type 2 poly-Bernoulli polynomials. They have also introduced unipoly functions attached to each suitable arithmetic function as a universal concept. Inspired by their work, in this paper, we introduce a new class of the Frobenius-Genocchi polynomials. We derive the diverse formulas and identities covering some summation formulas, derivative formula and correlations with Bernoulli polynomials and numbers, Stirling numbers of the both kinds, degenerate Frobenius-Genocchi polynomials and degenerate Frobenius-Euler polynomials. Moreover, by using the unipoly function as following Kim-Kim's work in <cite>Kim1</cite>, we consider degenerate unipoly-Frobenius-Genocchi polynomials and investigate some formulas and relationships with Daehee numbers, degenerate Frobenius-Genocchi numbers and Stirling numbers of the first kind. Finally, we obtain an Gaussian integral representation of the Frobenius-Genocchi polynomials in terms of the 2-variable Hermite polynomials.

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

Student-centered teaching practices such as active learning continue to gain momentum in college science education. Many instructors committed to these innovative practices transform their classroom beyond the standard lecture. Nevertheless, widespread implementation of these practices is limited because the learning benefits for students are often attained through increased instructional complexity to which many instructors cannot commit. When co-instructors are teaching the course, the level of commitment to building a student-centered classroom may be even more profound. For these reasons, new tools are needed to help instructors and co-instructors plan, organize, evaluate, and communicate their classroom innovations. Pathway modeling is a tool with potential to fill this gap. Unlike curriculum mapping -- which identifies academic content gaps, redundancies, and misalignments by examining a series of courses within a plan of study – course pathway modeling creates a visual map of a single course and reveals how teaching practices influence short-, mid-, and long-term student learning outcomes. This essay demonstrates how course pathway modeling can help co-instructors better represent the complexity of student-centered teaching practices. We include guides for creating course pathway models and discuss how this approach offers the potential to improve curricular design, course evaluation, student assessment, and communication between co-instructors.

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

Creative thinking is increasingly recognised as an essential ability that should be part of school curricula. Given the move towards online learning and assessment, we investigate whether mathematical creativity can be assessed at-scale in the Numbers game, an arithmetic game in Math Garden, a popular online math practice platform. In the Numbers game, a generalisation of the 24 Game, children are asked to figure out how to compute a target number using basic arithmetic operations and a given set of numbers. We argue that creative thinking is required when the search space is complex, and propose that the base-pattern, i.e., the sequence of the operations needed to solve a Numbers game item, indicates search space complexity. We then demonstrate that items with disordered base-patterns are more likely to require mathematical creativity to figure out. Specifically, our analysis shows that for items with only one solution sequence, those with disordered base-patterns are more difficult and take longer to solve compared to items with ordered base-patterns. For items where multiple solution sequences are possible, nine times out of ten children choose ordered over disordered base-patterns. We conclude that the Numbers game has potential for assessing mathematical creativity at-scale.

In this paper, we define new kind of Daehee numbers, the degenerate Daehee numbers of the third kind, using degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

The utilization of person centered care is highlighted as essential for health promotion, yet implementation has been inconsistent and multiple issues remain. There is a dearth of applied re-search exploring the facets of successful implementation. In this paper, a person centered wellbeing program spanning various groups is discussed outlining the central principles that have allowed for successful outcomes. The main data emerges from 10 years of pragmatic pre-post service evaluation. Measures of functional capacity and wellbeing were captured using validated measures. The method for this paper is a narrative exploration of the theory and practices that can explain the continual improvement the clinics have achieved over 10 years. Core principles relate to connecting with people, connecting through groups, and connecting with self. The operationalization and theoretical explanation of these principles is outlined alongside 10 years of data which shows sustained improvement in a range of outcomes. The discussion of these principles posits essential factors to prioritize to advance the implementation of person centered care in health promotion for long term conditions.

An ion exchange dialysis (IED) is used in the recovery of aluminium from residue. In this papers, the face-centered central composite design (FC-CCD) of the response surface methodology (RSM) and desirability approach is used for experimental design, modelling and process optimization of a counter flow IED system. The feed concentration, feed flowrate, sweep flowrate and sweep concentration are selected as the process variables, with the Al-transport across a Nafion 117 membrane as the target response. A total of 30 experimental runs were conducted with 6 center points. The response obtained was analysed by analysis of variance (ANOVA) and fitted to a second-order polynomial model using multiple regression analysis. The actual R2 and standard deviation of the model are 0.9548 and 0.2932 respectively. The influences of significant variables are plotted on 3D surface and contour plots. The designed variables were numerically optimized by applying the desirability function to achieve the maximum Al-transport. The optimised condition values were found to be feed concentration (1600 ppm), feed flowrate (61.76%), sweep flowrate (37.50%) and sweep concentration (0.75 N) for the 80% target response at 32hrs. Overall, the model can be used to effectively predict Al-recovery using the designed system

We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only, and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipoly- Fubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we introduce a new class of Laguerre-based Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Laguerre-based Hermite-Fubini numbers and polynomials by using generating functions.

Cerebellar mutism syndrome (CMS), also known as posterior fossa syndrome, is encountered in a subset of children who have undergone an operative procedure of the posterior cranial fossa, mainly involving the vermis, and the most frequent underlying pathology is proved to be medulloblastoma. The most common characteristics of this syndrome include an often transient, although protracted, language impairment, emotional lability, along with cerebellar, and brainstem dysfunction. Nevertheless, a significant number of patients experience persistent neurological deficits and lasting neurocognitive impairment. A lot of research and clinical studies have been performed in order to better delineate this syndrome. The main obstacles in our way to highlight all aspects of this syndrome were related with an inconsistent nomenclature, poorly defined diagnostic criteria, and uncertainty surrounding risk factors and etiology. Currently, there is a combination of diagnostic criteria that are regarded as prerequisites in order to establish the diagnosis of CMS. These include language impairment and emotional lability, as proposed by the international Board of the Posterior Fossa Society in their consensus statement. Several risk factors are recognized as implicated in the pathogenesis of this syndrome, including midline tumor location, diagnosis of medulloblastoma, younger age at diagnosis, and preoperatively established language impairment. A proposed etiology of CMS includes disruption of the cerebellar outflow tracts, the cerebellar nuclei, and their efferent projections through the superior cerebellar peduncle. Specific treatment for CMS is lacking, and it continues to be directed at symptom management. Our aim is to present a comprehensive narrative review of CMS etiology, diagnosis, risk factors, clinical presentation, and clinical management. Moreover, we attempt to recognize the most widely recognized priorities of the research community in order to expand our knowledge in the era of diagnostics, prevention, and therapeutic options for patients suffering from CM, or who are at risk for development of this syndrome.

This paper presents a new proof of the infinite nature of prime numbers. Grounded in fundamental Set theory and Analysis, the proof uses the method of contradiction to demonstrate the absurdity of assuming the set of all prime numbers to be finite.

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Z_p of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper we investigate some properties of square-free numbers. In particular, we study a function that counts the number of square-free numbers no larger than the given number n. We show that this function has asymptotics consistent with the predictions of the Riemann hypothesis and use this to obtain new and better estimates of the density of square-free numbers. Finally, we present a well-known generalization of the concept of a square-free number to monoids and research on square-free factorizations.

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

Mathematical modelling of bioprocesses has a long and notable history, with eminent contributions from fields including microbiology, ecology, biophysics, chemistry, statistics, control theory and mathematical theory. This richness of ideas and breadth of concepts provide great motivation for inquisitive engineers and intrepid scientists to try their hand at modelling, and this collaboration of disciplines has also delivered significant milestones in the quality and application of models for both theoretical and practical interrogation of engineered biological systems. The focus of this review is the anaerobic digestion process, which, as a technology that has come in and out of fashion, still remains a fundamental process for addressing the global climate emergency. Whether with conventional anaerobic digestion systems, biorefineries, or other anaerobic technologies, mathematical models are important tools that are used to design, monitor, control and optimise the process. Both highly structured, mechanistic models and data-driven approaches have been used extensively over half a decade, but recent advances in computational capacity, scientific understanding and diversity and quality of process data, presents an opportunity for the development of new modelling paradigms, augmentation of existing methods, or even incorporation of tools from other disciplines, to ensure that anaerobic digestion research can remain resilient and relevant in the face of emerging and future challenges.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper, we introduce a new class of $(p,q)$-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for $(p,q)$-Fubini polynomials associated with $(p,q)$-Bernoulli polynomials, $(p,q)$-Euler polynomials and $(p,q)$-Genocchi polynomials and $(p,q)$-Stirling numbers of the second kind.

In this paper, we investigate the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind, and derive some new and interesting identities and properties of those polynomials.

The main goal of the present paper is to construct some families of the Carlitz’s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤ_p. We then define the modified degenerate Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz’s q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz’s q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. Among which, the fuzzy preference relation is the reasonable one to represent preference relation by a fuzzy membership function. This paper studies the Nakamura’s and Kołodziejczyk’s preference relations. Eight cases of representing different level of overlapping between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of decomposition and intersection of two fuzzy numbers for the Nakamura’s and Kołodziejczyk’s preference relations of these test cases. The results indicate that the decomposition and intersection can affect the fuzzy preference relations, thereby the final total order relation of fuzzy numbers.

In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has been proposed, which led directly to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear DEs. The resulting polynomials can be used in non-relativistic and relativistic QM, particularly, in the case of the Schrödinger equation, and Dirac equations for an electron in a Coulomb potential field.

The study aimed to investigate how distributed leadership influences adolescents' social and emotional competence, considering the roles of student-centered instructional practices and teacher self-efficacy. Utilizing survey data from 7,246 Suzhou adolescents enrolled in the SESS program, the study employed multilevel structural equation modeling for analysis. The findings indicated a positive correlation between distributed leadership and adolescents' social and emotional competence, as evaluated by teachers. Moreover, distributed leadership was observed to exert an indirect impact on adolescents' self-assessment and teacher-rated social and emotional competence. This influence operated through several mediators: student-centered instructional practices, teacher self-efficacy, and the sequential mediating pathway of student-centered instructional practices → teacher self-efficacy. This study underscores the significance of supportive elements in fostering students' social and emotional competence. It offers valuable insights into the everyday practices of teachers, principals, and administrators. Additionally, it sheds light on how distributed leadership contributes to enhancing adolescents' overall social and emotional development.

Key thermophysical properties necessary for the successful design and use of CoCrFeNi alloy in thermophysical applications have been measured experimentally, and the results have been compared with literature values and results previously obtained for commercial Ni-Cr alloys and equiatomic CoCrFeNi alloy. In particular, the thermal diffusivity, coefficient of thermal expansion (CTE), and specific heat capacity were measured for the as-cast and homogenized equiatomic CoCrFeNi alloy over a temperature range allowing the thermal conductivity to be calculated up to 1173 K. The thermal conductivity and thermal diffusivity of the equiatomic CoCrFeNi alloy were found to deviate from monotonic behavior in the temperature range from 773 to 1100 K. Such a deviation was previously observed in the behavior of the temperature dependence of CTE and specific heat capacity of the equiatomic CoCrFeNi alloy. The non-linear behavior is primarily the result of order/disorder phenomena for the as-cast and homogenized sample, as well as additionally for the as-cast sample, non-equilibrium solidification under arc melting conditions. The measured data of thermophysical properties are provided for thermally differently treated samples, and it is shown that there is a difference in the behavior of the temperature dependences of CTE, thermal diffusivity, and heat capacity.

In this article, we discuss the concept of infinity (∞), infinitesimals □ , and a new mathematical operation that expands calculations into the realm of the infinite. This breakthrough allows for the computation of infinite numbers, leading to new possibilities for mathematical research and practical applications. By introducing infinite numbers, numerous advantages arise, such as accurately representing limits, divergences in integrals and infinite series, and extending calculations to infinity. This mathematical advancement opens doors to exploring uncharted territories, offering valuable insights and breakthroughs.

We developed some questions regarding twin prime conjecture by examining the bounded gaps between composite numbers and explained the relations between the formulas regarding composite numbers. Then we proved the twin prime conjecture to answer the new questions raised by proving the expression: \[ \liminf_{n\to\infty} (c_n_+_1 - c_n) \ge 10. \] In addition, we prove many different approaches have been and it has been proved that there must be a twin prime in the range $(p_k_2\cdot p_k_1, p^2_k_1)$, where $p \neq 2$.

This short technical note proves some observed novel corollaries of the Collatz Conjecture. For this, a unique ‘Collatz Representation’ of Hailstone numbers has been employed. After that, an extension of this corollary is explained using the fact that only certain differences in the last digit of the ‘Collatz Representation’ gives a whole number difference between numbers. A more general result is obtained. The final proof is for a corollary that some numbers the same number of steps to reach 1 if the Collatz Conjecture is true and all variables are whole numbers.

Starting with a positive odd integer $n$, apply a function $f(n)$ repetitively such that $f(n) = 3n + 1$ if $n \not\equiv 0 (\text{mod}\:2)$ or $f(n) = n/2$ if $n \equiv 0 (\text{mod}\:2)$. Let the sequence of all the odd integers obtained form an orbit $n, f^1(n) , f^2(n), \ldots, f^k(n), f^{k + 1}(n)$. The $k^{th}$ odd integer is written as $f^k(n) = (3^k + \mathbf{B(k)})/2^{\mathbf{z(k)}}$ where $\mathbf{B(k)}$ is the collection of terms without the coefficient $n$. We show that when $1 - \mathbf{B(k)}/(2^{\mathbf{z(k)}} n) \approx 1$, the condition for unbounded orbit is given by OEIS A022921. Further, if there exists at least one orbit for which $f^{k + 1}(n) = n$ when $(k + 1) < 3$, we show that there exists no orbit with $k + 1 > 3$ for which $f^{k + 1}(n) = n$. The final result is that the odd integers that can violate the Collatz conjecture are smaller than 5.

For each non-prime odd number as F=pq , if we consider m/n as an approximation for q/p and choose k=mn , then by proving some lemmas and theorems, we can compute the values of m and n. Finally, by using Fermat’s factorization method for F and 4kF as difference of two non-consecutive natural numbers, we should be able to find the values of p and q. Then we introduce two new and powerful sieves for separating composite numbers from prime numbers.

In standard construction of hyperrational numbers using ultra-power we assume that the ultra lter is selective. It makes possible toassign real value to any nite hyperrational number. So, we can consider hyperrational numbers with selective ultra lter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.