There is lacking information about the physiological response when conducting training of older men in a real-life setting. Seventy-six sedentary men aged 72.3 ± 0.6 (means ± SE; range: 63–92) years with a height, body mass and body mass index of 178.9 ± 0.8 cm; 92.1 ± 2.2 kg: 28.7 ± 0.6 kg/m2, respectively, were recruited to floorball (FG, n = 29), strength (SG, n = 38) or bowls (BG, n = 9) training 1 h twice a week in municipal activity centers and senior sport clubs. Subjects were tested at baseline, after 12 and 24 weeks. Twelve weeks of floorball and strength training lead to reduced blood Hb1Ac, body mass, fat mass, visceral and android fat. Further, SG had a decrease in gynoid fat and blood total and LDL cholesterol. Both groups decreased heart rate at rest. In SG, systolic and diastolic blood pressure were also reduced. FG increased markers for bone growth. FG and SG improved functional capacity. The improvements in FG and SG were maintained after 24 weeks. BG did not have any changes. In conclusion, older men conducting floorball or strength training twice a week in a real-life setting can improve functional capacity and a high number of health factors, whereas playing bowls does not lead to physiological changes.

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.

The electric network frequency (ENF), often referred to as the industrial heartbeat, plays a crucial role in the power system. In recent years, it has found applications in multimedia evidence identification for court proceedings and audio-visual temporal source identification. This paper introduces a ENF region classification model named UniTS-SinSpec within the UniTS framework. The model integrates the si-nusoidal activation function and spectral attention mechanism while also redesigning the model frame-work. Training is conducted using a public dataset on the open science framework platform (OSF), with final experimental results demonstrating that after parameter optimization, the UniTS-SinSpec model achieves an average validation accuracy of 97.47%, surpassing current state-of-the-art and baseline models. Accurate classification can significantly aid in ENF temporal source identification. Future re-search will focus on expanding dataset coverage and diversity to verify the model's generality and ro-bustness across different regions, time spans, and data sources. Additionally, it aims to explore extensive application potential of ENF region classification in preventing crimes such as telecommunications fraud, terrorism, and child pornography.

In this article, we present two topics related to mathematical economics. We offer an intriguing proposal for securing cost and production functions in the face of uncertainty and risk. We discuss the relationship between individual and social preferences. This is important because traditional revealed preference theory does not account for such factors.

The liver diseases are posing serious threat to human health associated with abusive use of allopathy medicines, especially in developing economies like Pakistan. The strengthening of detoxification system using natural products might be a suitable option to minimize the liver problems. In this current research, Zingiber officinale and Azadirachta indica were evaluated for their hepatoprotective role. For the purpose, 20 Sprague Dawley rats were equally divided into four groups i.e., negative control, paracetamol positive control, 2.5% ginger and 2% neem powder, respectively. During 42 days study, the levels of bilirubin and liver enzymes significantly elevated due to liver damage. Although, the values dropped (20-25%) in treatment group. Serum glucose and serum lipid profile increased in positive control group, but reduced (5-10%) in both ginger and neem groups. The serum protein profile declined in liver damage group, however, improved (~5%) in treatment groups. The hematological parameters slightly declined in liver damage group, but increased in ginger and neem groups. Moreover, the white blood cells differential count and renal parameters showed abnormal values in liver damage group, but improved after treatment. This study is promising and highlighted the hepato-protective role of neem and ginger that may open several avenues of research.

One of the most prevalent illnesses, diabetes does not directly result in patient mortality. But, it increases the risk of death. Any disease that may be predicted in its early stages can lessen its fatal effects while also enhancing the quality of the healthcare system. For the early-stage prediction of diabetes or such types of non-communicable diseases, we need the proper set of influential features. This research has developed a machine learning-based disease prediction model to identify the influential features for diabetes prediction and give a near-perfect classification accuracy. This model includes Min-Max normalization for data normalization, Isolation Forest (iForest) for outlier removal, Synthetic Minority Oversampling Technique (SMOTE) for oversampling, Random Forest based Recursive Feature Elimination (RFE-RF) test, Chi-Square test, and Minimum Redundancy Maximum Relevancy (mRMR) test based feature selection methods for identifying the influential features, and Support Vector Machine (SVM), K Nearest Neighbor (KNN), and Naive Bayes (NB) for the classification. The results clarify that the proposed model outperforms the previous models and studies. The SVM has attained an accuracy of 99.58% in classification using the five features chosen from the Chi-Square test. Lastly, SHAP, an explainable AI model, has been used to assess the classifier model’s performance. These selected features and the classifier model can be used for early-stage diabetes prediction.

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system, finite-time stable. The proof is based on a recursive design algorithm developed recently, to construct the virtual Hölder continuous, finite-time stabilizer as well as a C¹ positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz non-linear systems.

In today’s world of software development technologies, serverless cloud computing with microservices is a big deal. Microservices gives modularity, agility and scalability, while serverless allows to code without having to maintain any internal hardware infrastructure, so processes are streamlined and costs are reduced by 60% compared to traditional server based systems. In this paper we looked at the challenges of microservices and serverless architecture and how to address them. We analyzed existing architectures and proposed a conceptual framework that will give you a complete solution for building scalable and efficient cloud native applications that reduces costs by 30-50%, improves resource utilization by 15-20%. Cold start latency, a common issue in serverless computing is reduced by 25% using standardized optimization techniques. This paper also compares server based and serverless technologies on scalability, resource efficiency and cost. These findings will give you a serverless approach to cloud software development and supports the trend of architectural adoption so you can have more dynamic and efficient systems.

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

The intelligent morphing wing structure has the characteristics of flexibility, large deformation and adaptive change, which are different from those of the conventional wing. The motion law of the variable-camber leading edge is analysed by the numerical simulation method based on kinematics theory to solve the problem of verifying the motion function and structural strength of the variable camber wing. The driving deformation and aerodynamic load simulation of the variable camber leading edge are realised by designing a follow-up load test apparatus. Based on the numerical simulation of the mechanical boundary, the motion trajectory of the upper and lower wing loading points was predicted, and a multi-point cooperative control system was developed. Multi-sensor iteration was used to ensure the loading accuracy of the test control. The test results show that the average error of the deflection angle is 4.59% in the variable camber leading edge motion test. In the variable camber leading edge strength test, the average error of applied load magnitude and direction is 0.54% and 0.24%, respectively.

Image stitching is a crucial aspect of image processing. However, factors like perspective and environment often lead to irregular shapes in stitched images. Cropping or completion methods typically result in substantial loss of information. This paper proposes a method for rectifying irregularly images into rectangles using deformable meshes and residual networks. The method utilizes a convolutional neural network to quantify rigid structures of images. Choosing the most suitable mesh structure based on the extraction results, offering options such as triangular, rectangular, and hexagonal. Subsequently, the irregularly image, predefined mesh structure, and predicted mesh structure are input into a wide residual neural network for regression. The loss function comprises local and global, aimed at minimizing the loss of image information within the mesh and global target. This method not only significantly reduces information loss during rectification but also adapting to different images with various rigid structures. Validation on the DIR-D dataset shows this method outperforms state-of-the-art methods in image rectification.

In this paper the one-to-one correspondences and equivalences between ideals, primals, filters and grills are introduced. It is shown the local functions and the topological spaces induced by them are the same. From this point of view, the topological properties with respect to one topology can be derived from topological properties valid in the corresponding topology.

Suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A, using the Hausdorff outer measure, is measurable in the sense of Carathèodory; we want to find an extension of the expected value, w.r.t the Hausdorff measure, that's unique, finite and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of ℝᴬ. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x,fₖ*(x))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique, finite, and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.

The Adyar Creek, a backwater estuary at the mouth of the Adyar River on the Coromandel Coast of the Bay of Bengal, represents a critical ecosystem facing mounting pressure from urbanization. This study employed high throughput 16S rRNA gene sequencing to characterize bacterial communities within the riverine, estuarine, and coastal sediments of Adyar Creek. Proteobacteria was the dominant phylum across most samples, with proportions ranging from 39.65% to 72.09%. Notably, the estuarine environment exhibited a distinct taxonomic profile characterized by a significant abundance of Firmicutes (47.09% of the bacterial population). Distinct bacterial classes were observed across sediment types: Alphaproteobacteria (30.07% - 34.32%) in riverine sediments, bacilli dominated estuarine sediments (40.17%), and Gammaproteobacteria (15.71%-51.94%) in coastal sediments. The most significant environmental factors influencing bacterial community composition across these samples were pH, salinity, phosphate, and nitrate. LEfSe analysis identified specific genera within the estuary, including Bacillus (20.26%), unclassified Paenibacillaceae (12.87%), Clostridium (3.81%), Gailella (3.17%), Paenibacillus (3.02%), Massilia (1.70%), Paraburkholderia (1.42%), and Pantoea (1.15%), as potential biomarkers for habitat health. Functional analysis revealed elevated expression of genes associated with ABC transporters and carbon metabolism in the estuary, suggesting a heightened nutrient cycling capacity. Furthermore, co-occurrence network analysis indicated bacterial communities exhibit a strong modular structure with complex species interactions across the three sediment types. These findings highlight bacterial communities' critical role and their key drivers in estuarine ecosystems, establishing a baseline for further investigations into the functional ecology of these vulnerable ecosystems.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Evidence is emerging on the role of maternal diet, gut microbiota, and other lifestyle factors in establishing lifelong health and disease, which are determined by trans-generationally inherited epigenetic modifications. Understanding epigenetic mechanisms may help identify novel biomarkers for pregnancy-related exposure, burden, or disease risk. Such biomarkers are essential for developing tools for the early detection of risk factors and exposure levels. It is necessary to establish the exposure threshold to the fetus due to nutrient deficiencies or other environmental factors that can lead to clinically relevant epigenetic changes that modulate disease risks. This narrative review summarizes the latest updates on the roles of maternal nutrients (n-3 fatty acids, polyphenols, vitamins) and gut microbiota on the placental epigenome and its impacts on fetal brain development.

The challenging terrain and deep ravines characteristic of mountainous regions often result in slower path planning and suboptimal flight paths for Unmanned Aerial Vehicles (UAVs) when traditional meta-heuristic optimization algorithms are employed. This study investigates the application of an Improved Enhanced Snake Optimizer (IESO) for three-dimensional path planning in a simulated rugged mountainous terrain. The initialization process in the enhanced snake optimizer is refined by integrating the Chebyshev chaotic map. Additionally, a non-monotonic factor is introduced to modulate the "temperature," and a boundary condition is incorporated into the dynamic opposition learning mechanism. These modifications collectively reduce the likelihood of population convergence to local optima during optimization. The feasibility of IESO is validated through time complexity and global convergence analyses. Comparative simulation experiments benchmarked IESO against five state-of-the-art optimization algorithms across test functions and path-planning simulated scenarios. The experimental results indicate that the enhanced algorithm achieves superior optimization precision, faster convergence speeds, and improved quality in the planned trajectories by 30%.

Riemann’s functional equation is valid on the vertical line of s=1/2. Each side is a real-valued function. The Riemann’s Xi function is also a real-valued function along the vertical line of s=1/2. Through the holomorphic extensions of the Riemann zeta function, starting from the real-valued function at s=1/2 into both sides of sigma<1/2 and sigma>1/2, we can get two versions of the zeta functional equation, eq. (45). The key property of the scaling and rotational factors g(s) and g(1-s) behave as multiplicative inverses in the complex plane, eq. (48). It is deduced that the Zeta function also has multiplicative inverses, the symmetric point is at (1/2,0) in the complex plane. The moduli behave as a hyperbola. Especially, along the vertical line sigam=1/2, the amplitudes of both function g(s) and g(1-s) are equal to 1, its arguments have opposite signs. If sigma is not equal to 1/2, the amplitudes of zeta(s) and zeta(1-s) are not equal to each other, because of their multiplicative inversion relationship. It is deduced that the non-trivial zeros can only be on the vertical line of s=1/2. A gamma function vector field is given in Appendix B, and some moduli of gamma function at special points are given. Finally, another variation of the Zeta function is provided in an integral form in Appendix D. The asymptotes behave as a c₈ cyclic group for the large t values.

A general methodology for loan amortization under arbitrary discount functions is discussed. It is shown that it is always possible to uniquely define a scheme for constructing the loan amortization schedule with an arbitrary assigned discount function. It is also shown that even if the loan amortization is carried out from the sequence of principal payments and the sequence of accrued interest, the underlying discount function can be uniquely determined at the maturities corresponding to the installment payment dates. As a special case of the proposed approach, we derive the amortization method according to the law of simple interest.

Background: Codonopsis pilosula (CP) is one of the most popular Qi-invigorating herbal medicines and has been extensively used to promote health and vitality in China for a long time. C. pilosula polysaccharide (CPP) is the principal active components of CP, which is considered as the reason for CP widespread application. However, there are not revealed that CPP exert a Qi-invigoration effect by protecting mitochondria and/or improving mitochondrial function in the existing traditional Chinese medicine theories. Methods: Herein, we extracted CPP from C. pilosula and investigated the effects of CPP on energy metabolism and mitochondrial protection. Based on the mice chronic hypoxia model for imitating the “Qi” deficiency state of the human body in traditional Chinese medicine, which was administered with CPP by oral gavage daily for 10 days, mitochondrial permeability transition (MPT), lipid peroxidation product malondialdehyde (MDA) in brain, mitochondrial respiratory function, the levels of adenosine triphosphate (ATP), adenosine diphosphate (ADP) and adenosine monophosphate (AMP) in liver cells were assayed. Adenylate energy charge (AEC), total adenylate pool (TAP), ATP/ADP, and ATP/AMP ratios were calculated. Results: CPP could inhibit the formation of MDA in mice brains, decrease oxygen consuming rate and respiratory control ratio (RCR) of liver mitochondria, increase levels of ATP, TAP and AEC in liver cells under chronic hypoxia condition. Conclusion: Overall, CPP can invigorate “Qi” by improving mitochondrial energy metabolism and bioenergetic levels.

In this study, a cholera infection model with the bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to an relevant linearised system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show that the impact of regression speed and fluctuation intensity on stochastic systems.

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to find the sums over inverse powers of zeroes for the incomplete Gamma- and Riemann zeta- functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeroes of the Hurwitz zeta-function , including the sum over the inverse first power of its appropriately defined non-trivial zeroes. We also study some related properties of the Hurwitz zeta-function zeroes. In particular, we show that for any natural N and small real epsilon, when z tends to n=0, -1, -2… we can find at least N zeroes of zeta in the - vicinity of 0 for sufficiently small epsilon, as well as one simple zero tending to 1, etc.

Proof of the Riemann hypothesis is obtained using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.

It seems that natural compounds with bioactivity are the food constituents, present in the diets, that may provide protective effects on health due to their biochemical effects. The main benefits appear to be decline in oxidative stress and inflammation that are crucial in the pathogenesis of obesity, diabetes and metabolic syndrome. Apart from anti-inflammatory activity, these molecules also possess antifungal, and anti-infective preventative effects. Further evidence indicate that these bioactive molecules are the constituents, which are known to increase function of the foods. This review aims to emphasize that plant based diet is rich in phytochemicals such as antioxidants, vitamins, minerals, polyphenolics and flavonoids, amino acids and fatty acids, that are able to cause benefits via these compounds. These bioactive molecules can target mechanisms of oxidative stress and inflammation that are crucial in the management of metabolic syndrome. These antioxidant compounds appear to be essential component of plant based diets and are crucial for healthiness. Most investigators have proposed that bioactive compounds act as nutraceuticals which are able to regulate certain patho-physiological mechanisms responsible for metabolic diseases. The bioactive compoundspresent in the plant based diets are primary and secondary metabolites of nutritive and non-nutritive natural components generating health benefits by preventing or managing chronic disease or its symptoms. The bioactive compounds are known for their beneficial effects to health. Despite their small quantity in the diets, these bioactive compounds may trigger certain mechanisms that may cause alteration with improvement in health. It seems that most of the bioactive compounds co-existing in the plant-based diets, may be extremely healthful. However, if they are taken in excess, they can cause toxic effects. There is an urgent need to evaluate each bioactive compound to establish a beneficial quantity and a threshold of toxicity, while treating metabolic syndrome.

This paper is concerned with a hydraulic cylinder directly controlled by a variable speed fixed displacement pump. The considered configuration is referred to as a one-motor-one-pump (1M1P) motor-controlled hydraulic cylinder (MCC). A 1M1P MCC with a hydraulically driven passive load-holding function enabled by controlled system pressures is proposed. The proposed load-holding functionality works if there is a standstill command, a loss of power supply, or a hose rupture. Additionally, this paper conducts a comprehensive analysis of the proposed system's operation and load-holding function across four quadrants. Simulation results demonstrate the four-quadrant operation and good load-holding performance under the aforementioned scenarios. In conclusion, the proposed 1M1P MCC can be successfully used on practical applications characterized by overrunning loads and four quadrant operation.

The Polynomial coefficient function, from the complex hyperspace Cⁿ in itself, is the function Fⁿ that provides the coefficients of the polynomial (x - a₁)(x - a₂)... (x-a_n). A proof of the fundamental theorem of algebra is given where the surjectivity of this function is obtained from a Recurrent Coefficient Equation.

Objectives: To develop and validate the Communities Geriatric Mild Cognitive Impairment Risk Calculator (CGMCI-Risk), aimed at aiding community healthcare workers in the early identification of individuals at high risk of Mild Cognitive Impairment (MCI); Methods: Based on nationally representative community survey data, backward stepwise regression was employed to screen the variables, and logistic regression was utilized to construct the CGMCI-Risk. Internal validation was conducted using bootstrap resampling, while external validation was performed using temporal validation. The area under the receiver operating characteristic curve (AUROC), calibration curve, and decision curve analysis (DCA) were employed to evaluate the CGMCI-Risk in terms of discrimination, calibration, and net benefit, respectively; Results: The CGMCI-Risk model included variables such as age, educational level, sex, exercise, garden work, TV watching or radio listening, Instrumental Activity of Daily Living (IADL), hearing, and masticatory function. The AUROC was 0.781 (95% CI = 0.766 to 0.796). Calibration curve showed strong agreement, and the DCA suggested substantial clinical utility. In external validation, the CGMCI-Risk model maintained similar performance with an AUROC of 0.782 (95% CI = 0.763 to 0.801); Conclusions: CGMCI-Risk is an effective tool for assessing cognitive function risk within the community. It uses readily predictor variables, allowing community healthcare workers to identify the risk of MCI in older adults over a three-year span.

The fitting and modeling of skewed, complex, symmetric, and asymmetric datasets is an exciting research topic in many fields of applied sciences, notably lifetime, medical, and financial sciences. The Heavy tailed Nadarajah Haghighi model is introduced by compounding the heavy tailed family and Nadarajah Haghighi distribution in this paper. The so obtained model has three parameters accounting for the scale and shape of the distribution. The proposed distribution's fundamental characteristics such as probability density, cumulative distribution, hazard rate, and survival functions are provided, and several key statistical properties are established, and several entropy information measures are proposed. The estimation of model parameter is performed via the ML procedure. Further, different simulation experiments are conducted to demonstrate the performance of proposed estimator’s using some measure, like the average estimate, the average bias, and associated mean square error. Finally, we are applied our proposed model to analyze three different real datasets. In our illustration, we are compared the practicality of the recommended model with several well-known competing models.

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.

The anatomy of mature vertebrate hearts is marked by a looped design, which is roughly characterized by two components: (1), s-shaped (sigmoid) routing of the cardiac flow path(s), and (2,) bilaterally asymmetric (chiral) routing of the cardiac flow path(s). The looped heart design is regarded as a phylogenetically conserved feature among vertebrates and is thought to represent a significant determinant of the cardiac pumping function. It evolves during the embryonic period of development by a morphogenetic process called “cardiac looping”. During the past decades, remarkable progress has been made in the uncovering of the genetic, molecular, and biophysical factors contributing to cardiac looping. Thereby, a special focus was directed on the evolution of a bilaterally asymmetric (chiral) heart configuration, which is said to play a central role in the determination of the alignments and separations of the pulmonary and systemic flow paths in the multi-chambered hearts of tetrapods. In contrast to the recent progress made in uncovering of the biological driver of cardiac looping, our present knowledge about the functional consequences of cardiac looping is at an immature state. This article provides an overview and discussion of the currently available information about the looped design of vertebrate hearts and its functional implications.

Traditional path planning algorithms have drawbacks in unmanned vehicle path planning, such as low efficiency, unsuitability for dynamic environments, and high computational complexity. To address these issues, this paper proposes a dynamic path planning method for unmanned vehicles that combines the A* algorithm with an improved Dynamic Window Approach (DWA). Firstly, a collision risk function is incorporated into the DWA algorithm to evaluate scores based on the braking distance corresponding to the current vehicle speed and the distance to obstacles, determining whether timely braking or obstacle avoidance is feasible. The earlier the braking or obstacle avoidance, the higher the score, thereby enhancing the evaluation algorithm of the DWA. Secondly, the key points extracted by the A* algorithm are used as temporary target points for the DWA algorithm, reducing the eight search directions to five. Finally, the key points of the A* algorithm’s path are set as the temporary end points for the DWA algorithm, integrating the two algorithms to plan a smoother curved path that can avoid unknown and dynamic obstacles. Simulation data demonstrates that compared to traditional artificial potential field methods, RRT, and Dijkstra’s algorithm, the new fusion algorithm reduces time by 11.71 seconds, a decrease of 6.9%, and shortens path length by 3.093 meters, a reduction of 4.16%, while increasing average linear velocity by 2.9%. The proposed algorithm ensures smoother paths while also making paths more rational and efficient

Background: 3D virtual reality (VR) Motionless Imagery Exercise through Avatar (MIEA) was developed to improve elderly’s psychological (cognitive and emotional) and physical health. This study observed the impact of this program on the cognitive functions and emotional well-being and the electrophysiological changes in older adults. Methods: This study was conducted using a randomized controlled trial design. The participants included 38 adults, 65 to 84 years of age, who were physically mobile and neurologically healthy. Among them, 19 participants were assigned to the experimental group, who underwent exercises using elderly avatars in a VR virtual exercise program. The other 19 participants were assigned to the control group who underwent the VR program. Both groups engaged in the VR intervention for 20 minutes three times a week, over six weeks. Results: The cognitive functions, including attention restraint, working memory, and phonemic fluency, of the experimental group showed significant improvements compared to the control group. Regarding emotional well-being, the experimental group showed greater self-efficacy. The electrophysiological changes revealed similar patterns to exercise-induced reactions in electromyography of the experimental group after intervention. Conclusions: These findings confirm that participation in 3D VR MIEA enhances cognitive functions and physical self-efficacy among the elderly. Therefore, 3D VR MIEA can be utilized in the rehabilitation therapy of patients or elderly with limited physical activity.

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, the symmetrical deformation equation of elastic toroidal shells is successfully transferred into a well-known equation, namely Heun's equation of ordinary differential equation, whose exact solution is obtained in terms of Heun's functions. The computation of the problem can be carried out by symbolic software that is able to with the Heun's function, such as Maple. The Gauss curvature of the elastic toroidal shells shows that the internal portion of the toroidal shells has better bending capacity than the outer portion, which might be useful for the design of metamaterials with toroidal shells cells. Through numerical comparison study, the mechanics of elastic toroidal shells is sensitive to the radius ratio. By slightly adjustment of the ratio might get a desired high performance shell structure.

This longitudinal study aimed to analyze the effects of a team rowing-based training program on physical fitness and anthropometric parameters in female breast cancer survivors (n=40; 56.78±6.38 years). Participants were divided into two groups: one rowed in fixed seat rowing (FSR) boats (n=20; 56.35±4.89 years) and the other in sliding seat rowing (SSR) boats (n=20; 57.20±7.7 years). Both groups engaged in two 75-minute sessions per week for 24 weeks. Significant improvements were observed in both groups in resting heart rate (FSR: -10.65 bpm; SSR: -8.45 bpm), heart rate at the beginning of the 6-minute walk test (6MWT) (FSR: -10.7 bpm; SSR: -11.25 bpm), and heart rate at the end of the test (FSR: -13.85 bpm; SSR: -20.35 bpm). Blood pressure improved significantly in both diastolic blood pressure (FSR: -12.35 mmHg; SSR: -19.25 mmHg) and systolic blood pressure (FSR: -13 mmHg; SSR: -16.95 mmHg). Additionally, both groups increased the distance covered in the 6MWT (FSR: +63.05 m; SSR: +93.65 m). These results suggest that a rowing training program is a viable and safe activity for female breast cancer survivors, improving cardiac function, blood pressure, and cardiorespiratory capacity, particularly in sliding seat boats.

The Marchenko method is a data-driven way which makes it possible to calculate Green's functions from virtual points in the subsurface by the reflection data at the surface, only requiring a macro velocity model. This method requires collocated sources and receivers. However, in practice, subsampling of sources or receivers will cause gaps and distortions in the obtained focusing functions and Green's functions. To solve this problem, this paper proposes to integrate sparse inversion into the iterative Marchenko scheme. Specifically, we add sparsity constraints to the Marchenko equations and apply sparse inversion during the iterative process. Our work not only reduces the strict requirements on acquisition geometries, but also avoids the complexity and instability of direct inversion for Marchenko equations. This new method is applied to a two-dimensional numerical example with irregular sampled data. The result shows that it can effectively fill gaps of the obtained focusing functions and Green's functions in the Marchenko method.

In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.

This study explores the inelastic doubly differential transverse momentum spectra of the primary charged particles, (π + + π − ), (K + + K − ) and (pp̄) as a function of underlying event √ (UE) at s = 13 T eV . The particle production is measured on the basis of different angular regions like toward, transverse and away, elucidated with respect to the direction of leading particle of an event. To study the thermal freeze-out parameters, the non extensive Tsallis distribution function is used, to extract the temperature Tef f , and chemical potential µ, which provides a basis to explain the QCD matter. The Tsallis distribution function clearly describes transverse momentum spectra in pseudorapidity region of |η| < 0.8. The best fit corresponds to minimum χ2 /ndf value for the transverse momentum distribution. It is observed that effective temperature Tef f changes from away to towards and forward region.

Indoleamine 2,3-dioxygenase 1 (IDO1) and IDO2, originated from gene duplication before vertebrate diver-gence. While IDO1 has a well-defined role in immune regulation, the biological role of IDO2 remains unclear. Discovered in 2007, IDO2 is located near the IDO1 gene. Because of their high sequence similarity, IDO2 was initially thought to be a tryptophan (Trp)-degrading enzyme like IDO1. Differently from what expected, IDO2 displays extremely low catalytic activity toward Trp. Nevertheless, many studies, often contradictory, tried to demonstrate that IDO2 modulates immune responses by catabolizing Trp into kynurenine, a hypothesis un-convincing and, let’s say, linked to the first life of IDO2. Here, we review its “new life”, the one in which IDO2 has a functional role beyond Trp metabolism. The IDO2’s evolutionary persistence across species, despite be-ing almost inactive as an enzyme, suggests it has some relevant biological importance. IDO2 expression in human normal cells is poor, but significant in various cancers, with two prevalent SNPs. Overall, the com-parison of IDO2 to IDO1 as a Trp-degrading enzyme may have led to misunderstandings about IDO2’s true physiological and pathological roles. New insights suggest that IDO2 might function more as a signaling molecule, particularly in cancer contexts, and further studies could reveal its potential as a target for cancer therapy.

Body-centered cubic bismuth (Bi) is considered an enticing pressure marker, and therefore it is highly desirable to command its accurate equation of state (EOS). However, significant discrepancies are noted among the previous experimental EOSs. In the present work, the EOS up to 300 GPa is theoretically obtained by direct solving the partition function via a direct integral approach (DIA). The calculated results nearly reproduce the hydrostatic experimental measurements below 75 GPa, and the deviations from the measurements gradually get larger with increasing pressure. Based on the ensemble theory of equilibrium state, DIA works with high precision particularly in high-pressure conditions, so the hydrostatic EOS presented in this work is expected to be a reliable pressure standard.

Abstract: Background: Restless Legs Syndrome /Willis-Ekbom disease (RLS/WED) has occasionally -but not consistently- been associated with cognitive, and most notably, language and executive impairment. The present study was conducted to investigate the cognitive trajectories of older individuals with RLS/WED. Methods: Participants were drawn from the randomly selected, older (>64 years), population-based HELIAD cohort. Individuals without dementia, with available neuropsychological evaluations at baseline and follow-up were considered for potential eligibility. A comprehensive assessment examining five principal components of cognition (memory, visuo-spatial ability, attention, executive function, and language) was administered to the partici-pants. Generalized estimating equations analysis was used to examine the unadjusted and adjusted (for critical factors and covariates) effects of RLS/WED on cognition over time. Results: A total of 1003 predominantly female, older (72.9 ±4.9 years) partici-pants, with follow-up evaluations after a mean of 3.09 ±0.85 years and without demen-tia at baseline and follow-up were included in the present study. Among them, 81 were diagnosed with RLS/WED at baseline. Global cognition, memory, attention, executive and visuo-perceptual skills did not differ between those with and without RLS/WED. However, the RLS/WED group performed worse on language at baseline by 0.249 of a standard deviation, while demonstrating a mitigated language decline over time, by 0.063 of a standard deviation. The unadjusted models yielded similar results. Conclu-sions: Our findings were indicative of a baseline language disadvantage among older individuals with RLS/WED, but the initial discrepancy tends to dissolve over time.

The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modelling and forecasting. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive explicit solution. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega function. The present manuscript reports new analytical results and numerical routines suitable for parametric estimation of the SIR model. The manuscript introduces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The numerical approach is exemplified with data from the European Centre for Disease Prevention and Control (ECDC) for several European countries in the period Jan 2020 -- Jun 2020.

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

The aims of this study are to measure the technical efficiency (TE) of Cambodian household’s rice production and trying to determine its main influencing factors using the stochastic frontier production function. The study utilized primary data collected from 301 rice farmers in three selected districts of Battambang by structured questionnaires. The empirical results indicated the level of household rice output varied according to differences in the efficiency of production processes. The mean TE is 0.34 which means that famers produce 34% of rice at best practice at the current level of production inputs and technology, indicates that rice output has the potential of being increased further by 66% at the same level of inputs if farmers had been technically efficient. Furthermore, between 2013-2015 TE of household’s rice production recorded -14.3% decline rate due to highly affected of drought during dry season of 2015. Moreover, evidence reveals that land, fertilizer, and pesticide are the major influencing input factors of household’s rice production, while disaster, education of household head, family size and other crops’ cultivated area are core influencing factors decreasing TE. Conversely, the main influencing factors increasing TE are irrigated area, number of plot area and sex of household head.

The aims of this study are to measure the technical efficiency (TE) of Cambodian household’s rice production and trying to determine its main influencing factors using the stochastic frontier production function. The study utilized primary data collected from 301 rice farmers in three selected districts of Battambang by structured questionnaires. The empirical results indicated the level of household rice output varied according to differences in the efficiency of the production processes. The mean TE is 0.34 which means that famers produce 34% of rice at best practice at the current level of production inputs and technology, indicates that rice output has the potential of being increased further by 66% at the same level of inputs if farmers had been technically efficient. Furthermore, between 2013-2015, TE of household’s rice production recorded -14.3% decline rate due to highly affected of drought during dry season of 2015. Moreover, evidence reveals that land, fertilizer, and pesticide are the major influencing input factors of household’s rice production, while disaster, education of household head, family size and other crops’ cultivated area are core influencing factors decreasing TE. Conversely, the main influencing factors increasing TE are irrigated area, number of plot area and sex of household head.

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein.

In this paper we investigate the differential topological properties of a subclass of singular space--subcarteisan spaces. First, we get further result on partition of unity for differential spaces. Second, we establish the tubular neighborhood theorem for subcartesian spaces with constant structural dimension. Third, we generalize the concept of Morse functions on smooth manifolds to differential spaces. For subcartesian space with constant structural dimension, we provide examples of Morse functions; With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, we prove that smooth bounded functions on this space can be approximated by Morse functions; We study the infinitesimal stability of Morse functions on subcartesian spaces. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

This study investigates the efficacy of an integrated rehabilitation program that combines vestibular and knee-focused therapies to improve proprioceptive function, balance, and motor control in patients suffering from both Benign Paroxysmal Positional Vertigo (BPPV) and knee joint conditions. The study involved 59 participants diagnosed with both vestibular dysfunction and knee impairments, who were enrolled in a comprehensive rehabilitation program. Vestibular interventions included specific maneuvers such as the Epley maneuver and Brandt-Daroff exercises, aimed at repositioning otoliths and reducing vertigo symptoms, while balance retraining exercises were implemented to enhance stability and prevent falls. Additionally, TECAR therapy was used for its deep tissue heating and regenerative benefits, and specific proprioceptive exercises targeting the knee joint were conducted to improve joint position sense and reduce the risk of falls. The intervention resulted in significant improvements in proprioceptive accuracy, as evidenced by enhanced joint position sense measurements and increased scores on the Dynamic Gait Index (DGI). These findings suggest that the dual-targeted rehabilitation approach not only addresses the individual impairments associated with vestibular dysfunction and knee pathology but also effectively improves overall functional outcomes by tackling the complex interplay between these conditions. The study further highlights the importance of a multidisciplinary approach in rehabilitation, where addressing vestibular and musculoskeletal dysfunctions simultaneously can lead to better patient outcomes. The integrated rehabilitation strategy presented in this study offers a comprehensive approach to managing complex clinical cases, ultimately contributing to improved quality of life and functional independence in patients affected by concurrent vestibular and knee joint issues.

In this paper, the time-resolved model of photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e. structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed.

Reconstructing 3D indoor scenes from 2D images has always been an important task in computer vision and graphics applications. For indoor scenes, traditional 3D reconstruction methods suffer from the lack of surface details, poor reconstruction of large flat surface textures and areas with uneven lighting, and many falsely reconstructed floating debris noises in the reconstructed models. We add adaptive normal priors to the neural implicit reconstruction process to optimize the network, and improve the accuracy of volume density prediction by adding regularization terms to the neural radiation field to constrain the volume density obtained by weight distribution, and learn a smooth SDF surface from the network to obtain an explicit mesh model. Experiments show that the method proposed in this paper outperforms the state-of-the-art methods on ScanNet, Hypersim, and Replica datasets.

Background: Upper extremities’ function impairments (UEFI) and gait impairment are frequently reported even in the early stages of Multiple Sclerosis (MS) disease. These motor deficits can negatively affect the quality of life (QOL) in MS patient. Exercise has been repeatedly introduced as a beneficial adjuvant therapy for MS, attenuating a wide range of physical and psychological symptoms without any side effects or increased risks of relapse or exacerbation in disease symptoms.Aim: This paper investigates the effect of multi-function swing suspension training (MFSST) program on upper and lower extremities’ function (ULEF), and QOL in MS women according to expanded disability status scale (EDSS) score.Design: Experimental method with a pre-test and post-test design with control group. Setting: The department of Sports Injuries and Corrective Exercise at the Shahid Bahonar University of Kerman.Population: Forty-seven MS women.Methods: Eligible subjects who matched the selection criteria were randomly assigned into three groups as follow: i. Intervention group A (EDSS of 2 – 4), ii. Intervention group B (EDSS of 4.5 – 6.5), and iii. Control group (EDSS of 2 – 6.5). The intervention groups have been trained by the SET program for a duration of 8 weeks. At baseline and after the intervention, the upper extremities’ function (UEF) and the lower extremities’ function (LEF) were assessed by 9-hole peg test (9HPT) and the timed 25-foot walk (T25FW) test, respectively. Moreover, participants’ QOL was measured using the Short Form questionnaire (SF-36).Results: A significant difference was found between the pre-test and post-test variables of walking speed and upper extremity endurance in intervention groups, and between the intervention groups and control group (P < 0.05). Compared to control group, both intervention groups showed improvements in all subscales of QOL, except emotional role limitation score.Conclusions: In summary, these results show that the present protocol of MFSST could be effective in improving ULEF and QOL in MS patients, showing the improvements after 4^th week. Moreover, this protocol was more useful for patients with an EDSS score of more than 4.5. Therefore, it seems that exercise therapy should be considered as a beneficial clinical intervention approach to improve ULEF and QOL in MS patients.

Background: Alzheimer's Disease (AD) is a progressive and incurable condition that prompts the exploration of interventions to slow its advancement and enhance the quality of life. The cortical brain waves could have a role in the process of executive function deterioration, increasing dementia symptoms. Objective: To interrogate the effects of physical exercise, and non-invasive brain stimulation, both in electrophysiological behavior of the brain, the executive functions, and dementia symptoms in older adults with AD. Methods: One hundred and sixteen older adults were grouped (n=22 each): GAE (AD, PMED+NIBS), GCP (AD, PMED), GCN (AD, NIBS), GCS (control, no intervention). Binaural beats were used as a non-invasive brain stimulation, physical exercise, or both, were applied during 16 weeks. Pre/post measurements included Alpha Band, and SMR Rhythms, executive functions, and dementia symptoms. Results: GCS showed no significant changes (p>0.05). GCN and GCS slightly enhanced inhibitory control (p>0.05) and working memory (p>0.05). In contrast, GCP and GAE improved executive function (p

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

Background Cytomegalovirus (CMV) is a major pathogen that cause remarkable rate of morbidity and mortality, especially in immunocompromised patients. It is important to find risk factors associated with CMV viremia. We studied the differences in CMV seropositivity in relation to liver function biomarkers in male and female Saudi population in an attempt to understand the variation in the CMV seroprevalence with sex and find the risk factor to develop liver dysfunction or hepatocellular carcinoma. Material and subjects: The CMV- IgG and IgM were screened in serum samples of 150 non- A-G hepatities patients with elevation of liver profiles (ALT, AST, ALP and GGT) and categorized as males and females. Samples were collected from different general hospitals and polyclinic in KSA from March 2014 to June 2015. A correlation between CMV seropositivity measured with both antibodies and liver enzymes were tested. Receiver operating characteristics (ROC) analysis and multiple regressions were done for the obtained data. Results: Our study shows that females had much higher IgG and IgM compared to age-matching males. A significant correlation between both antibodies and liver enzymes (AST, ALT) was recorded. Less significant correlation of both IgG and IgM with GGT was also observed. Receiver operating characteristics (ROC) analysis revealed that both IgG and IgM can be used as excellent predictive markers for CMV infection as both recorded 100% specificity and sensitivity together with area under the curve of 1 in males and females. Multiple regression analysis ascertain the correlation between both antibodies as dependent variables and liver enzymes as independent variables with ALT being the most affected enzyme with CMV seropositivity especially in females. Conclusion:he data discussed above This study shows that CMV is capable of initiating and accelerating liver dysfunction in both sexes. The high seroprevalence in females at reproductive age is especially important as they can transmit the virus to their developing fetus. Prevention of CMV infection in young girls 11-14 years old, through counseling on hygiene or possible future vaccination, may lead to a decrease of congenital CMV infections with the concomitant risk of developing liver dysfunction or hepatocellular carcinoma. Keywords: Cytomegalovirus, Alanine transaminase, Aspartate transaminase Alkaline phosphatase, γ-Glutamyltranspeptidase, liver function.

In the context of evolving perceptions of sexuality, particularly within the realm of health and disability, this study investigates the impact of multiple sclerosis (MS) on female sexual function and quality of life. A quantitative study involving 130 female MS patients aged 35 to 50 was conducted, employing measures such as The Female Sexual Function Index (FSFI), The Multiple Sclerosis Intimacy and Sexuality Questionnaire-19 (MSISQ-19), and The Fatigue Severity Scale (FSS). Results indicate a significant association between greater sexual dysfunction and poorer quality of sex life, alongside the correlation of increased fatigue with diminished sexual satisfaction. Factors such as education level, time of diagnosis, and treatment regimen also influenced outcomes. Findings underscore the importance of supportive interventions, including counseling, to enhance the quality of sex life for women with disabilities, particularly those with MS.

The skin, as the body's largest organ, is vital for protecting against environmental stressors, regulating temperature, and preventing water loss. Maintaining a healthy skin barrier is essential for overall skin health and function. We examined the potential of a mixture of five traditional Korean herbal extracts—Cimicifuga racemosa, Paeonia lactiflora, Phellodendron amurense, Rheum rhaponticum, and Scutellaria baicalensis— referred to as herbal medicine composite 5 (HRMC5) for enhancing skin health. High-performance liquid chromatography identified 14 bioactive compounds, including flavonoids, phenolic acids, anthraquinones, and alkaloids, known for their antioxidant, anti-inflammatory, and antimicrobial effects. The extract showed concentration-dependent effects on cell viability and provided UV protection at 6.25 g/L. It also significantly promoted wound healing, similar to the efficacy of epidermal growth factor. Furthermore, the mixture exhibited anti-inflammatory activity by downregulating Cox2 expression and upregulated key skin barrier proteins, including Involucrin, Claudin 1, and Collagen Type 1. These findings highlight the potential of this herbal mixture for addressing skin barrier dysfunction, inflammation, and facial redness, supporting its use in developing natural skincare products. Future studies should investigate the mechanisms behind these effects and assess its clinical applications.

In this paper, the feasibility and potential assessment (PA) of wind power in six different areas MehalMeda, Eneware, DebreBerhan, AlemKetema, GundoMekel, and Majet are investigated. The data is collected from the National Metrological Agency (NMA) in Addis Ababa and from the districts of the case study areas and analyzed using MATLAB and HOMER software. Using the data collected at 10 meter height, wind direction using a wind rose, the wind frequency distribution, and wind turbine characteristics have been analyzed. Moreover, to obtain a more realistic wind potential study, the wind speed at a 10 meter height is extrapolated to a height of 50 meter in order to determine wind speed, wind power density, and wind energy density. The average wind speed of the zone at 10m and 50m heights are 1.58 m/s and 1.99 m/s, 3.06 m/s and 3.85 m/s, 4.75 m/s and 5.98 m/s, 2.29 m/s and 2.88 m/s, 1.01 m/s and 1.27 m/s and 4.07 and 5.12m/s at Alemketema, Debreberhan, Eneware, Gundemeskel, Majete and Mehalmeda respectively. The extrapolated result shows that the maximum wind speed and power densities at a height of 50 meter for Eneware, MehalMeda and Debre Berhan are 10.4 m/s, 689 W/m2, 8.9m/s, 429 W/m2 and 6.3m/s, 152.2 W/m2 respectively. The energy densities at a height of 50 meter for AlemKetema, DebreBerhan, Eneware, GundoMekel, Majete, and MehalMeda are 181.06 kWh/m2*yr , 1332.83 kWh/ m2*yr, 6078.91 kWh/ m2*yr, 857.76 kWh/ m2*yr, 58.78 kWh/ m2*yr, and 3939.42 kWh/ m2*yr respectively. As per the international standard for wind power and wind speed classification, considering the extrapolated height, Eneware, MehalMeda, and DebreBerhan have feasible wind potential for both small-scale and large-scale wind farm implementation, whereas Majete, AlemKetema, and GundoMeskel of the North Shewa Zone in the Amhara region have poor wind potential.

In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.

This paper proposes training improvement methods of artificial neural networks (ANN) trajectory predictors. First, a dynamic power system time-series trajectory is split into several different segments to simplify the original ANN training problem. Moreover, the time-derivative of the trajectory is included to obtain an augmented loss function. Compared to previous studies which mainly focused on increasing the prediction accuracy, the aim of these novel techniques is to reduce the computational burden where the ANN output performance is still acceptable. The effectiveness of the developed methods is validated based on the WSCC three-machine nine-bus and IEEE 39-bus system models. The mean absolute error (MAE) and trajectory prediction results are analyzed, in which the numbers of neurons, hidden layers, and training epochs are constrained during the ANN training process. Rotor-angle difference between generators and the system frequency are investigated as the dynamic trajectories of the power system models. It is revealed that when the ANN architecture and epochs are constrained, the MAE results can be reduced, and the ANN training results can better reflect the original trajectory using the improvement approaches.

The objective of this study is to examine a nonparametric estimate , using the kernel approach, of the conditional distribution function of a scalar response variable that is given a random variable whose values take place in a separable real Hilbert space. The observations will be dependent on one another in a quasi-associated fashion. The pointwise practically perfect consistencies with rates of this estimator are established by us under some broad conditions. The study’s major objective is to investigate the convergence rate of the proposed estimator and its application in the convergence rate and asymptotic normality of the hazard function. The asymptotic normality of the developed estimator is established precisely. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data.

We developed a novel hybrid approach for solving global optimization, computer science, bio-medical and engineering real life applications that is based on the coupling of the Whale Optimizer and Sine Cosine Algorithms via a surrogate model. We relate the whale optimizer algorithm to balance between the exploitation and the exploration process in the proposed method. There exist confirmed techniques for searching approximate best optimal solutions, but our algorithm will further guarantee that such numerical and statistical solutions satisfy physical bounds of the standard and real life functions. Our experiments with the benchmark, bio-medical, computer science and engineering real life problems have illustrated the advantages of using a newly hybrid approach based on mixing Whale Optimizer and Sine Cosine algorithms. It holds considerable potential for reducing execution time for solving standard and real life problems and at the same time improving the quality of the solution.

Dry eye syndrome (DES) has become increasingly prevalent due to the prolonged use of digital devices in modern society. Individual supplementation with lutein or docosahexaenoic acid (DHA) demonstrated potential benefits for eye health. This study aimed to investigate the effects of lutein plus DHA complex supplementation on DES, visual function, and memory in healthy individuals. Thirty-four participants, aged 40~60 years, received lutein/DHA complex supplements (containing 10 mg free-form lutein, 200 mg DHA, 30 mg red algae extract (astaxanthin), phosphatidylserine, and vitamin A) for 12 weeks. Ophthalmological examinations and memory tests were carried out, and a questionnaire was administered at weeks 0, 6, and 12 of the study. After 12 weeks of the intervention, the tear break-up time had significantly increased, and intraocular pressure (IOP) had significantly decreased. This suggests that lutein/DHA complex supplementation may reduce the IOP and offer potential benefits for DES, as indicated by improved tear stability.

The objective of this study is to evaluate the effect of constitutive equations on the prediction accuracy for springback in cold stamping with various deformation modes. In this study, two types of yield functions—Hill’48 and Yld2000-2d—were considered to describe yield behavior. Isotropic and kinematic hardening models based on the Yoshida–Uemori model were also adopted to describe hardening behavior. Various material tests (such as uniaxial tension, tension- compression, loading-unloading, and hydraulic bulging tests) were carried out to determine the material parameters of the models. The obtained parameters were implemented in the finite element (FE) simulation to predict springback, and the results were compared with experimental data. U-bending and T-shape drawing were employed to evaluate the springback prediction accuracy. Obviously, the springback prediction accuracy was greatly influenced by constitutive equations. Therefore, it is important to choose appropriate constitutive equations for accurate description of material behaviors in FE simulation.

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

In the paper, the author finds bounds of the sine and cosine along a circle on the complex plane in terms of two double inequalities for the norms of the sine and cosine along a circle on the complex plane.

Arthropods, the most abundant group in ecosystems, play crucial roles in human production. Soil physicochemical properties reflect soil health and are linked to arthropods through cascading effects. Gravel-sand mulching, commonly practiced in dry farming regions, initially aids in water retention and temperature regulation. However, long-term cultivation can lead to land degradation and habitat fragmentation. This study aimed to investigate field systems using multiple comparison, linear, and nonlinear correlation analyses to explore the effects of different tillage durations on soil properties and arthropod diversity under gravel-sand mulching. Results indicate that arthro-pod diversity was lower while the fields maintained better soil fertility than the desert steppe. Both soil fertility and arthropod diversity declined significantly with increased years of cultivation. Mantel tests revealed no significant response of desert steppe arthropods to the altered soil properties of gravel-sand mulched fields. However, the Generalized Additive Model analysis indicated significant linear correlations between arthropod occurrence and total nitrogen (positive), available phosphorus, and water content (negative). Arthropod abundance showed significant nonlinear correlations with total potassium, alkali hydrolyzable nitrogen, and organic matter (OM), with OM changes having the most pronounced impact. Thus, gravel-sand mulching can maintain soil fertility over time but at the expense of local arthropod diversity.

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

In order to solve the problem of feature extraction of underwater acoustic signals in complex ocean environment, a new method for feature extraction from ship radiated noise is presented based on empirical mode decomposition theory and permutation entropy. It analyzes the separability for permutation entropies of the intrinsic mode functions of three types of ship radiated noise signals, and discusses the permutation entropy of the intrinsic mode function with the highest energy. In this study, ship radiated noise signals measured from three types of ships are decomposed into a set of intrinsic mode functions with empirical mode decomposition method. Then, the permutation entropies of all intrinsic mode functions are calculated with appropriate parameters. The permutation entropies are obviously different in the intrinsic mode functions with the highest energy, thus, the permutation entropy of the intrinsic mode function with the highest energy is regarded as a new characteristic parameter to extract the feature of ship radiated noise. After that, the characteristic parameters, namely, the energy difference between high and low frequency, permutation entropy, and multi-scale permutation entropy, are compared with the permutation entropy of the intrinsic mode function with the highest energy. It is discovered that the four characteristic parameters are at the same level for similar ships, however, there are differences in the parameters for different types of ships. The results demonstrate that the permutation entropy of the intrinsic mode function with the highest energy is better in separability as the characteristic parameter than the other three parameters by comparing their fluctuation ranges and the average values of the four characteristic parameters. Hence, the feature of ship radiated noise can be extracted efficiently with the method.

In this paper, a new approach to the sliding-mode control of single-phase inverters under linear and non-linear loads is introduced. The main idea behind this approach is to utilize a non-linear, flexible and multi-slope function in controller structure. This non-linear function makes the controller possible to control the inverter by a non-linear multi-slope sliding surface. In general, this sliding surface has two parts with different slopes in each part and the flexibility of the sliding surface makes the multi-slope sliding-mode controller (MSSMC) possible to reduce the total harmonic distortion, to improve the tracking accuracy, and to prevent overshoots leading to undesirable transient-states in output voltage which are occurred when the load current sharply rises. In order to improve the tracking accuracy and to reduce the steady-state error, an integral term of the multi-slope function is also added to the sliding surface. The improved performance of the proposed controller is confirmed by simulations and finally, the results of the proposed approach are compared with a conventional SMC and a SRFPI controller.

An inverse-square probability mass function (PMF) is at Newcomb-Benford Law’s (NBL) root and ultimately at the origin of positional notation and conformality. Under its tail, we find information as harmonic likelihood, leading to the global NBL ruled by the base B. Under its tail, we find information as logarithmic likelihood, leading to the local (fiducial) NBL ruled by the radix R. In the framework of bijective numeration, we prove that the set of Kempner’s series conforms to the global NBL and that the local NBL is length- and position-invariant. The global Bayesian rule multiplies the correlation between numbers, s and t, by a likelihood ratio that is the NBL probability of bucket [s;t) relative to B’s support. To encode the odds of quantum j against i locally, we multiply the prior odds Pr(B;j)/Pr(B;i) by a likelihood ratio that is the NBL probability of bin [i;j) relative to R’s support. This two-factor structure is recurrent under arithmetic operations. A particular case of Bayesian data produces the algebraic field of "referential ratios". The cross-ratio, the central tool in conformal geometry, is a ratio of referential ratios. A one-dimensional coding source reflects the harmonic external world into its logarithmic coding space, the ball {x∈Q| abs(x)< 1-1/B}. The source’s conformal encoding function is y=logR(2x-1), where x is the observed Euclidean distance to an object’s position. The conformal decoding function is x = ½ (1 + ry). Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality.

This paper presents a theorem in factorial functions with the sum of any two nonnegative integers that is equal to multiple of the product of factorial of the same two nonnegative integers.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

We show succinctly a natural origin of the beautiful Gamma function in light of the WJ distribution, linking to quite distinct fields.

The closed forms of the non-resonant thermonuclear function in the Maxwell-Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The Behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their further work in this area.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

Hypertrophic cardiomyopathy (HCM) is a common form of cardiomyopathy in cats, and heart failure occurs as diastolic dysfunction progresses. HCM in cats is broadly classified as non-obstructive and obstructive hypertrophic cardiomyopathy, depending on the presence or absence of outflow tract obstruction. Measurement of the intraventricular pressure differences (IVPD) using Color M-mode (CMM) has attracted attention as a reliable diastolic index as it correlates with catheterization, the gold standard for assessment of diastolic performance. Since, IVPD is affected by the size of the heart, the intraventricular pressure gradient (IVPG) index, which is unaffected by heart size, is used by dividing IVPD be LV length. In the present study, CMM IVPG was used to non-invasively assess diastolic impairment in cats with obstructive hypertrophic cardiomyopathy. This study was conducted on ten control cats and 18 cats in the HCM group. Although no severe left atrial enlargement was observed in the HCM group, the Basal IVPG was significantly increased in the HOCM group compared to the control group. While IVPD typically suggests impaired diastolic function and reduced ventricular compliance, the significant increase observed in the HOCM group compared to controls may suggest an indirect elevation in left atrial pressure, likely secondary to left ventricular outflow tract obstruction. The increase in IVPG in HOCM, as shown in this study, is a pathological effect of left ventricular outflow tract obstruction that cannot be detected by conventional echocardiographic indices, and evaluating IVPG is useful for evaluating cardiac function from a different perspective than conventional methods.

We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.

Kim(2015) introduced the degenerate Cauchy numbers of the second kind and gave some identities of them. In this paper, by using the modified polyexponential functions which are introduced by Kim-Kim(2019), we define the degenerate poly-Cauchy polynomials of the second kind and investigate some properties of them. Furthermore, we consider the degenerate unipoly-Cauchy polynomials of the second kind and discuss some properties of them.

This paper is devoted to the study of multi-dimensional integral transform with Fox H-function in the kernel in weighted spaces integrable functions in the domain R+n with positive coordinates. Mapping properties such as the boundedness, the range, the representations of the considered transformation are established.

In economics, especially econometrics, fixed-effects are widely recognized as powerful tools for heterogeneity analysis of longitudinal data sets. However, the existing research pays more attention to such as the variable selection, parameter estimation and dimension reduction on linear relationships with too many assumptions while nonlinear relationships are ubiquitous. In this paper, an extension fixed-effects model is proposed on the curve_fit nonlinear least squares. First, with the help of expert knowledge, the series regressions between independent variables including time and dependent variable is estimated on each series data, and the parameters are estimated by curve_fit. Similarly, the coeffection between independent variables including time and dependent variable is also initially estimated on the whole panel data. Second, the distance between regressions is defined, and then the coeffect function between the series regressions is obtained on the initial co-effect with curve_fit. Third, the time-effects and individual fixed-effects for each series data in the panel data are estimated on the differences of series regressions and co-effect function. Additionally, the effectiveness of the proposed method is varified on the synthetic data.

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.

Financial system is essentially chaotic and unstable if there is not any external inputs. By means of Lyapunov function method, design of switching law, novel fuzzy assumption, $L^p$ estimation technique and Laplace semigroup theory, the author presents the boundedness and LMI-based (globally) asymptotical input-to-state stability criteria of financial systems. Particularly, the globally asymptotical stability in the meaning of switching implies that when the time $t$ is big enough, the dynamic of any subsystem must approach its unique equilibrium point. Besides, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless. So the stability in the meaning of switching proposed in this paper is suitable and appropriate. Numerical examples illuminate the effectiveness of the obtained results.

We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our analysis focuses on examining the iteration sequence (or composition) of the Collatz function and its reduced variant, which serves as an analog to the inverse function, to scrutinize the validity of the Collatz conjecture. To substantiate this conjecture, we incorporate binary strings into an algebraic formula that captures the essence of the Collatz sequence. By this means, we transform discrete powers of 2 into continuous counterparts, ultimately culminating in the smallest natural number, 1. Consequently, the sequence generated through infinite iterations of the Collatz function emerges as an eventually periodic sequence, thereby validating an enduring 87-year-old conjecture.

We use binary string to represent a natural number and show the composite procedure of odd-number and even-number functions, thus we propose full binary directed tree to represent the set of natural numbers and give another partition the natural number set to three sets: pure odd, pure even and mixed number. For the Collatz conjecture we make use of the parity of a natural number to analyse the sequence of iteration (or composite) of Collatz function and reduced Collatz function analog to the inverse function. We give tabular and binary strings to the algebra expression to state the sequence of Collatz, this is the key topic to proof the conjecture: the discrete powers of 2 can be changed to ultimately continuous powers of 2, finally get to pure even number and to the smallest number 1. The sequence created by the infinite iterations of the Collatz function becomes the ultimately periodic sequence if any natural number is the beginning value, proving the conjecture that has been held for 87 years.

It is argued that if the non-unitary measurement transition, as codified by Von Neumann, is a real physical process, then the ‘probability assumption’ needed to derive the Second Law of Thermodynamics naturally enters at that point. The existence of a real, indeterministic physical process underlying the measurement transition would therefore provide an ontological basis for Boltzmann’s Stosszahlansatz and thereby explain the unidirectional increase of entropy against a backdrop of otherwise time-reversible laws. It is noted that the Transactional Interpretation (TI) of quantum mechanics provides such a physical account of the non-unitary measurement transition, and TI is brought to bear in finding a physically complete, non-ad hoc grounding for the Second Law.

In recent decades, with the ever-growing scale of video data, near-duplicate videos continue to emerge. Data quality issue caused by near-duplicate videos is becoming more and more prominent, which has affected the application of normal videos. Although current studies on near-duplicate video detection can be helpful to uncover data quality issues for videos, they still lack a process of automatic merging for the video data represented by high-dimensional features, which are difficult to automatically clean the near-duplicate videos to improve data quality for video datasets. At present, there are few studies on near-duplicate video data cleaning. The existing studies have the sensitive problems of video data orderliness and clustering initial center under a condition that prior distribution is unknown, which seriously affect the accuracy of near-duplicate video data cleaning. To address the above issues, an automatic near-duplicate video data cleaning method based on a consistent feature hash ring is proposed in this paper. First, a residual network with convolutional block attention modules, a long short-term memory deep network, and an attention model are integrated to construct an RCLA deep network with the multi-headed attention mechanism to extract spatiotemporal features of video data. Then, a consistent feature hash ring is constructed, which can effectively alleviate the sensitivity of video data orderliness while providing a condition of near-duplicate video merging. To reduce the sensitivity of the initial cluster centers to results of near-duplicate video cleansing, an optimized feature distance-means clustering algorithm is constructed by utilizing a mountain peak function on a consistent feature hash ring, which can implement automatic cleaning of near-duplicate video data. Finally, experiments are conducted based on a commonly used dataset named CC_WEB_VIDEO and a coal mining video dataset. Compared with some existing works, simulation results demonstrate that the performance of the proposed method.

The tunable work function of titanium dioxide (TiO₂) nanoparticles of various sizes was measured using the Kelvin Probe Force Microscopy (KPFM) technique. The analysis of the contact potential difference (CPD) across TiO₂ nanoparticles of different sizes revealed a clear relationship between nanoparticle size, surface roughness, and work function. The study observed work function values ranging from 4.49 eV to 4.57 eV for particle sizes between 3 nm and 85 nm, which were higher than those of bulk TiO₂, likely due to quantum confinement effects. Additionally, electrostatic force microscopy (EFM) measurements showed significant charge-trapping behavior within TiO₂ nanoparticles under different applied bias voltages. The optical analysis also revealed a quantum confinement-induced band gap of 3.35 eV, larger than that of bulk TiO₂, and distinct photoluminescence peaks at 383 nm and 403 nm, corresponding to near-band-edge excitonic emissions and defect-related states. By analyzing the conduction band (CB) and valence band (VB) of TiO₂ nanoparticles using KPFM, the CB and VB positions were calculated based on work function data and bandgap. The results indicated that as particle size increases, both CB and VB energies shift slightly towards lower values, where smaller nanoparticles exhibit larger band gaps and higher work functions.

Machine unlearning, the process of selectively forgetting or removing the influence of specific data points from a machine learning model, is increasingly important for privacy and compliance with regulations like the GDPR. This paper explores the concept of exact unlearning, focusing on its implementation in models trained using convex and non-convex functions. Convex functions, due to their well-behaved optimization landscapes, lend themselves to efficient unlearning through methods such as inverse optimization, duality-based approaches, and incremental learning. In contrast, non-convex functions, common in deep learning models, present more complex challenges due to their multiple local minima and high-dimensional parameter spaces. Techniques like checkpoint-based retraining, gradient inversion, and meta-learning are discussed as viable, though computationally expensive, methods for non-convex exact unlearning. The paper also highlights real-world applications in fields such as finance and healthcare, where exact unlearning can enhance privacy and security without compromising model performance. Finally, it outlines key challenges and future research directions, particularly the need for more efficient unlearning algorithms in non-convex settings and the development of secure, adversarial-resistant methods for sensitive data removal.

This paper provides the straight proof for the Riemann Hypothesis.The proof has been formulated with the modified version of Riemann Xi function ξ(s) , that shows the relationship with upper incomplete gamma function.

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$.

In history, theta function identities have been studied extensively and deeply. In this paper, applying the residue theorem of an elliptic function, we obtain a parameterized theta function identity. Then, by selecting the appropriate parameter, we derive some famous Jacobi identities, as well as other relatively simple theta function identities.

In this paper, an analytical form of the Unit Step Function (or Heaviside Step function) is presented. This important function constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics. In particular, Heaviside Step Function is performed in a very simple manner by the use of a finite number of standard operations, as the summation of six inverse tangent functions. The novelty of this work when compared with other analytical expressions, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Beta function, Logistic function. Besides, it is neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

A new set of basis functions is presented. The foundation in mathematics is established. To reconstruct a signal, it is compared to the Fourier basis functions. The M-basis functions' potential applications are also presented.

I want to find an extension of the Hausdorff Measure and Fractal Measure which allows me to compute a unique, satisfying average for functions defined on non-fractal, measurable sets in the sense of Carathedory which has no gauge function.

Spectral calibration consists of the calibration of wavelengths and the measurement of the instrument's spectral response function (SRF). Unlike conventional slits, the absorbed aerosol sensors (AAS) are used as a slit homogenizer, in which the spectral response function (SRF) is not a conventional Gaussian curve. To be more precise, the SRF is the convolution of the slit function of the spectrometer, the line spread function of the optical system, and the detector response function. The SRF of the slit homogenizer is a flat-topped multi-Gaussian function. Considering the convenience of fitting, a super-Gaussian function, which looks like a similar distribution to the flat-topped multi-Gaussian function, is used to fit the measured data in a spectral calibration. According to the results, the SRF’s shapes, due to the Earth port, resemble a Gaussian curve with a flatted top could be concluded, which contains an FWHM of 1.78–1.82 nm for the AAS. The results show that the correlation coefficients are about 0.99, which proves that the fitting function could better characterize the SRF of the instrument.

In this work, we investigate the behaviour of relativistic quantum oscillator under the effects of Lorentz symmetry violation determined by a tensor $(K_F)_{\mu\nu\alpha\beta}$ out of the Standard Model Extension. We analyze this relativistic system under an inverse radial electric field and a constant magnetic field induced by Lorentz symmetry violation. We see that the presence of Lorentz symmetry breaking terms modified the energy spectrum of the system, and a quantum effect arise due to the dependence of the linear charge density on the quantum numbers of the system

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π

Let $f(t)$ be a characteristic function. The question on infinite divisibility of $g_{2k}(t)=f^{(2k)}(t)/f^{(2k)}(0)$ is considered. There are given the condition for that function not to be infinite divisible. Some examples of infinite divisibility of $g_{2k}(t)$ are given.