Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$

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

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

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.

The Solvency II regulatory framework specifies procedures and parameters for determining solvency capital requirements (SCRs) for insurance companies. The proposed standard SCR calculations involve two steps. The Value–at–Risk (VaR) of each risk driver is measured and, in a second step, all components are aggregated to the company’s overall SCR, using the Standard Formula. This formula has two inputs: the VaRs of the individual risk drivers and their correlations. The appropriate calibration of these input parameters has been the purpose of various Quantitative Impact Studies that have been conducted during recent years. This paper demonstrates that the parameter calibration for the equity–risk module—overall, with about 25%, the most significant risk component—is seriously flawed, giving rise to spurious and highly erratic parameter values. As a consequence, an implementation of the Standard Formula with the currently proposed calibration settings for equity–risk is likely to produce inaccurate, biased and, over time, highly erratic capital requirements.

This paper presents a novel architecture for detecting mathematical formulas in document images, which is an important step for reliable information extraction in several domains. Recently, Cascade Mask R-CNN networks have been introduced to solve object detection in computer vision. In this paper, we suggest a couple of modifications to the existing Cascade Mask R-CNN architecture: First, the proposed network uses deformable convolutions instead of conventional convolutions in the backbone network to spot areas of interest better. Second, it uses a dual backbone of ResNeXt-101, having composite connections at the parallel stages. Finally, our proposed network is end-to-end trainable. We evaluate the proposed approach on the ICDAR-2017 POD and Marmot datasets. The proposed approach demonstrates state-of-the-art performance on ICDAR-2017 POD at a higher IoU threshold with an f1-score of 0.917, reducing the relative error by 7.8%. Moreover, we accomplished correct detection accuracy of 81.3% on embedded formulas on the Marmot dataset, which results in a relative error reduction of 30%.

The diagnosis and management of cow's milk allergy (CMA) is a topic of debate and controversy. We aimed to compare the opinions of expert groups from the Middle East (n=14) and the European Society of Pediatric Gastroenterology, Hepatology and Nutrition (ESPGHAN) (n=13). These Expert groups voted on statements developed by the ESPGHAN group and published in a recent position paper. The voting outcome was compared. Overall, there was consensus amongst both groups of experts. Most experts agreed that symptoms of crying, irritability and colic, as single manifestations, do not suggest CMA. They agreed that amino-acid-based formula (AAF) should be reserved for severe cases (e.g. malnutrition and anaphylaxis) and that there is insufficient evidence to recommend a step-down approach. There was no consensus on the statement that a cow’s milk-based extensively hydrolysed formula (eHF) should be the first choice as a diagnostic elimination diet in mild/moderate cases. Although the statements regarding the role of hydrolysed rice formula as a diagnostic and therapeutic elimination diet were accepted, 3/27 disagreed. The votes regarding soy formula highlight the differences in opinion in the role of soy protein in CMA dietary treatment. Generally, soy-based formula is seldom available in the Middle East region. All ESPGHAN experts agreed that there was no added value of probiotics, prebiotics, and synbiotics to the efficacy of elimination diets, whereas 3/14 of the Middle East group thought there was sufficient evidence. Therefore, there is not a consensus on the optimal diagnostic and therapeutic elimination diet for CMA, but the first-choice elimination diet and the stepwise management approach reveal some differences.

In this paper we use the Mellin convolution theorem, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil for other arithmetic functions different from the Von-Mangoldt function.

This mathematical observation deals with the development of a completely new expression [(S/4D) × √ {(S₁²-D²)(D²-D₁²)}] to calculate the area of a trapezoid system. Because of the difficulties of the previous methods to find out the area of a trapezoid (AOT), the proposed formula has thus indicated the best way to determine the AOT more precisely and easily beyond limitation, successfully. It is obvious that the derived formula is helpful to determine AOT more easily and precisely rather than existing method. And this developed theorem for the trapezoid system is observed for the first time.

We approximate the trigonometric function sine and cosine on the interval . This analysis provides two formulas to approximate sine and cosine. At first, we try to derive the formula which involves a square root, and then we derive another formula that does not require any use of a square root. Nevertheless, after deriving the procedure which requires no square root, we further try to increase its accuracy and then derive another formula that approximates trigonometric functions more accurately on the interval [0, pi/2]. So, this analysis provides mainly two types of procedures. One uses square roots, whereas the other does not. We also focus on ensuring the accuracy of these trigonometric functions in the interval [0,pi/2]. This accuracy analysis is portrayed using the graph. This graph shows the difference between the values generated by the functions defined here and the actual value of these functions. So, these graphs also indicate the error of these functions on that interval. Finally, we compare our approximation with the approximation formula of the 7th-century Indian Mathematician Bhaskara I.

This paper is another applied article based on the author's "Unified Theory of Force". The main work of this paper is as follows: (1) Reinterpretation of the Lorentz contraction effect and rigorous proof that the Lorentz contraction is a physical phenomenon based on the conservation of energy and momentum under non-completely symmetrical relative motion effects. This paper proves that the Lorentz contraction effect not only occurs in the direction of motion but also simultaneously in any direction. (2) For the first time, we attempted to explain the physical meaning of the Planck energy formula in theory and associate the Planck energy formula with the Lorentz contraction effect, which can help us to have a deeper understanding of the true physical meaning behind the Planck energy formula. At the same time, we also made reasonable theoretical predictions about the basic structure of photons.

Actively controlling the camber angle to improve energy efficiency has recently gained interest due to the importance of reducing energy consumption and the driveline electrification trend that makes cost-efficient implementation of actuators possible. To analyse how much energy that can be saved with camber control, the effect of changeing the camber angles on the forces and moments of the tyre under different driving conditions should be considered. In this paper, Magic Formula tyre models for combined slip and camber are used for simulation of energy analysis. The components of power loss during cornering are formulated and used to explain the influence that camber angles have on the power loss. For the studied driving paths and the assumed driver model, the simulation results show that active camber control can have considerable influence on power loss during cornering. Different combinations of camber angles are simulated, and a camber control algorithm is proposed and verified in simulation. The results show that the camber controller has very promising application prospects for energy-efficient cornering.

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

Copper, which can potentially be a highly toxic agent, is an essential nutrient due to its role as a co-factor for cuproenzymes and participation in signaling pathways. In mammals, the liver is a central organ that controls copper turnover throughout the body: copper absorption, distribution, and excretion. In ontogenesis, there are two types of copper metabolism: embryonic and adult, which maintain the balance of copper in each of these periods, respectively. In the liver cells, these types are characterized by specific expression patterns and activity levels of the genes encoding ceruloplasmin, which is the main extracellular ferroxidase and copper transporter and proteins mediating ceruloplasmin metalation. In newborns, the molecular-genetic mechanisms responsible for copper homeostasis and the ontogenetic switch from embryonic to adult copper metabolism are highly adapted to milk ceruloplasmin as a dietary source of copper. In the mammary gland cells, the level of ceruloplasmin gene expression and the alternative splicing of its pre-mRNA govern the amount of ceruloplasmin in milk, and thus, the amount of copper absorbed by the newborn is controlled. In the newborns, absorption, distribution, and accumulation copper are adapted to milk ceruloplasmin. In the newborns, which are not breast-fed at the early stages of postnatal development, the control for alimentary copper balance is absent. We tried to focus on the neonatal consequences of a violation of the balance of copper in the mother / newborn system. Although there is still much to be learned, the time to pay attention to this problem came because the neonatal misbalance of copper may provoke the development of copper related disorders for future life.

This paper is the third application-oriented paper based on the "unified theory of forces" and also addresses the shortcomings of previous research in fully explaining the origin and nature of the weak force. The main conclusions of this paper are as follows: (1) It is argued that neutrinos are generated by the gravitational field, and their generation mechanism is similar to the radiation of photons from electron orbital transitions. Particles undergoing orbital transitions in a gravitational field will emit neutrinos. (2) We rigorously derive the energy formula for neutrinos and demonstrate that the energy formula for neutrinos does not comply with the Planck energy formula. Neutrinos with higher energy have longer wavelengths and are more easily observed in experiments.

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

An approximate calculation of the spatial characteristics on finite range is required, so one quantitative continuum represents the accumulation of infinite great quantities is artificially divided it into smaller and camparable parts in which calculus operation can be applied .This operation is defined as Theorem 1 in which infinity is not involved, there is a camparable finity is constantly (forever) approaching and not reaching infinity, and only staying within a finite range. Theorem 1 can exist in this paper as a new mathematical basis for physics. Because the essence of all physical quantities is size comparison, and the size comparison relation of matter can only be space/time, so relation formula space/time is the only expression of the concept of matter, all physical quantities are applicable to this expression, each different physical quantity is a multi-dimensional representation of this expression. A new mass energy formula is aslo derived from this paper.

In this study, we investigate spectral structure of conformable Sturm-Liouville problems and with this end, we obtain representation of solutions under different initial conditions and asymptotic formulas for eigenfunctions, eigenvalues, norming constants and normalized eigenfunctions. Consequently, we prove the existence of infinitely many eigenvalues. Also, we compare the solutions with graphics with different orders, different eigenvalues, different potentials and so, we observe the behaviors of eigenfunctions. We give an application to the α-orthogonality of eigenfunctions and reality of eigenvalues for conformable Sturm-Liouville problems defined by [15] in the last section.

BACKGROUND: It has been known for decades that breastfeeding leads to a lower risk of asthma, respiratory infections, or metabolic syndrome at school age. In addition, evidence is now accumulating on the influence of breast milk on the shape, volume, or function of the heart and lungs. Within this field of research into the effects of breast milk on the structure of the heart and lungs, we have set out to analyze the differential electrocardiographic characteristics of schoolchildren who were once breastfed. METHOD: Observational cross-sectional study including 138 children aged 6 or 12 consecutively presenting to a well-child clinic between May and December 2022. Inclusion criteria: the ability to perform reproducible ECG records, the feasibility of weighing and measuring patient, and breastfeeding data collected from birth. RESULTS: Using the 40º cut-off value for the mean P-wave axis among schoolchildren, 76% of never breastfed children in our sample have a P-wave axis in a more vertical position than the mean as compared to 58% of ever-breastfed children (OR:2.25;95%CI:3.13-1.36); there was no other significant difference between infant feeding groups in somatometric characteristics or ECG parameters, CONCLUSION: We found a significant difference of the mean values of the P-wave axis between never and ever breastfed children. Although this report should be approached cautiously, these findings add to the renewed interest in discerning developmental interventions to improve cardiovascular health.

Background. Increasing evidence demonstrate that concentration of protein in infant formula >1.9g/100Kcal with high levels of insulinogenic aminoacids is associated with accelerated weight gain, increased fat mass accumulation and risk of adiposity. Purpose of this study was to conduct a systematic review to determine the metabolic effects in infants feed with infant formula optimized in protein. Methods. Systematic review was conducted according PRISMA Statement. RCTs with one intervention group (infant formula with 1.6-1.9gr of protein/100Kcal) and at least one comparative control group (infant formula with >1.9gr of protein/100Kcal) were included. Standardized mean differences (SMD), through random model were calculated. Results. 15 RCT were included. Optimized protein in infant formula was associated with less gain of BMI at 24 months of follow-up (SMD -0.25, IC95% -0.36 to -0.13, p 0.01) and less fat mass accumulation (SMD -0.68, IC95% -0.98 to -0.37, p 0.01). Optimized protein was also associated to less gain of weight, weight/age Z-score, weight/length Z-score, BUN (mmol/dL) and IGF1 (ng/ml). No effect on length/age Z-score was observed. Conclusions. Robust evidence showed optimized protein (1.6gr/100Kcal to 1.9gr/100Kcal) in infant formula is associated with metabolic benefits in infants with less weight gain, BMI and fat mass accumulation.

The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

Trailing oil is the tail section of contamination. There are two main reasons for the formation of trailing oil, one is the effect of laminar flow boundary layer, the other is the outflow of the preceding batch remained in the dead-legs. In the batch transportation of refined oil, under the action of viscous force, the preceding batch forms laminar boundary layer near the pipe wall and stays on the pipe wall, resulting in the phenomenon of contamination trailing and formation of trailing oil. When oil passes through the valve chamber of the oil transportation station, dead-leg will be formed. Due to gravity and convection diffusion, preceding batch flowing from dead-legs will form trailing oil in the pipeline. The phenomenon of trailing oil exists in the process of batch transportation, which will have an effect on the quality of oil. In this paper, Reynolds time-averaged method is used to simulate turbulence.Computational Fluid Dynamics(CFD) software is used to simulate different flow rates and bypass lengths to obtain contamination-related experimental data.Matlab software is used to perform multi-nonlinear regression for the oil substitution time, the length of the bypass and the flow rate. The formula for calculating the length of the trailing oil produced by the dead-leg is obtained. The modified formula for calculating the length of the contamination is obtained by combining the existing formula for calculating the length of the contamination.

Based on the 5.7-magnitude earthquake that stroke Songyuan (China) and 172 groups of liquefaction data collected in mainland China, the hyperbolic liquefaction discriminant formula originally proposed by Sun Rui was revised, and a new formula for the liquefaction of sand was put forward. Groups of data derived from the Bachu earthquake in Xinjiang and an earthquake that occurred in New Zealand (47 and 195 groups, respectively) were used to carry out a back-judgment test, then, the results were compared with those of the existing standard method. Overall, the results showed that the new formula for hyperbolic liquefaction discrimination compensates for the conservative liquefaction discrimination of the older formula; moreover, it has a good applicability to different intensities, groundwater levels, and the deep sand layer of the Songyuan site, reflected by a more balanced success rate. Therefore, combining the existing liquefaction discrimination methods and the research results of discrimination, it is necessary to establish a suitable regional identification method through the continuous accumulation of liquefaction data and expanding database.

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

Bell’s correlation formula is based on a classical Kolmogorov probability theory. It is tested in experiment with an estimate correlation. The latter is the raw product moment (rpm) correlation that is classical Kolmogorov based as well. We note that the famous Bell inequalities are derived from Bell’s correlation formula. The problem is that the hypothesis: “is it possible that in experiment Bell’s correlation is equal to quantum”, can only found to be false. This is not because the data generating process does not produce the required data. Or that the inequalities sufficiently show that the data cannot violate the inequalities. It is because the statistics of the Bell experiment is flawed. The rpm correlation in experiment cannot be equal to the quantum value in principle. This is because it requires the classical Kolmogorov rpm correlation to be not Kolmogorov.

Rail impedance directly affects the transmission performance of track circuit . Considering the condition of earth stratification, for the difficult to calculate the rail impedance due to the semi-infinite integration interval and the oscillation of the integrand by using the Carson formula, The truncation method is proposed to divide the impedance formula is divided into definite integral and tail integral. The integral is approximated by the spline function, and the tail integral is calculated by using the exponential integral and Euler formula. Based on it, the rail impedance calculation formula of track circuit is obtained. The electromagnetic field model of track circuit with earth stratification is simulated by finite element method, and the correctness of the method is verified. Based on the formula, the influence of current frequency, soil depth and conductivity on rail impedance is studied. The relative error between the calculated results of rail impedance and the simulation results of finite element is within 5%. It can be seen that the formula has high accuracy and correctly reflects the law of rail impedance variation with current frequency, soil depth and resistivity. It provides a reliable reference for the theoretical calculation of rail impedance of track circuit.

In this paper is discussed sediment transport as a mechanical process that characterises a natural stream or channel flow regime. The objective of experimental work presented in this paper is to recall and to give another prospect of well-known Meyer-Peter and Müller approach for estimation of Shield’s number (θ_c,θ) in laboratory conditions, and calibration of dimensionless MPM number (A). For this purpose two different experiments are conducted, during the first experiment water amount flushed on the flume and bed slope was changed simultaneously until equilibrium state is achieved, meanwhile is estimated the critical Shield’s number (θ_c). While, during the second experiment, water amount was kept constant, only bed slope of flume was continuously tilted, meanwhile sediment, discharge and Shield’s number (θ) was determined for given hydraulic conditions. In addition calibration of dimensionless MPM number (A) was performed, where several iteration were considered until for (A=3.42), sediment discharge measured become almost equal with sediment discharge computed by using MPM formula. After these experiments, is concluded that MPM formula can be used also for other certain initial condition and similar procedure may be adopted to calibrate the dimensionless MPM number (A) .

Inspired by Dirac's large number hypothesis (LNH), Zeldovich proposed a relation between the cosmological constant (CC), the mass of a proton, the Planck constant and Newton's gravitational constant. Zeldovich's formula suggests that the substance which is responsible for CC may interact with elementary particles. Since Newton's gravitational constant and the Planck constant appear in Zeldovich's formula, these interactions may be related to gravitational phenomena and quantum phenomena. Following this clue, we propose a liquid droplet model of electron. Applying the Laplace's equation, Zeldovich's formula of CC is derived. Weinberg proposes a relation between the mass of a pion, the Planck constant, the Hubble constant, Newton's gravitational constant and the velocity of light in vacuum when discussing cosmological models with a varying constant of gravitation. Weinberg pointed out that this relation may not merely be regarded as a meaningless random combinations of these fundamental constants in physics. The Weinberg's relation is derived based on the critical mass density formula in the Friedmann model of the universe and Zeldovich's formula.

Our research is in the tangent space of a point on a 4-dimensional Riemannian manifold. Besides the positive definite metric, the manifold is endowed with a tensor structure of type (1,1), whose fourth power is minus the identity. Both structures are compatible and they define an indefinite metric on the manifold. With the help of the indefinite metric we determine a circle in different 2-planes in the tangent space on the manifold, we also calculate the length and area of the circle. On a smooth closed curve such as a circle, we define a vector force field. Further, we obtain the circulation done by the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analogue of the well known Green’s formula in the Euclidean space.

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

In this paper, by using the q-integer, we introduce a new generalization of Fibonacci and Lucas sedenions called q-Fibonacci and q-Lucas sedenions. We present some fundamental properties of these type of sedenions such as Binet formulas, exponential generating fuctions, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity.

A new generalization of extended beta function and its various properties, integral representations and distribution are given in this paper. In addition, we establish the generalization of extended hypergeometric and con uent hypergeometric functions using the newly extended beta function. The properties these extended and con uent hypergeometric functions such as integral representations, Mellin transformations, dif- ferentiation formulas, transformation and summation formulas are also investigated.

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

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.

An extension of the Stoney formula for the case of a back side metallized 8” silicon taiko wafer has been developed, in the elastic regime, within the frame of the theory of elasticity. A good correlation between the calculated warpage, determined by the stress released by a given back side metallization (BSM), and the corresponding experimental warpages of the same thick metal layers deposited on an 8” silicon taiko wafer provides evidences of the correctness of the developed theory. This development suggests the possibility to extend this approach to the case of 8” taiko wafers based on a wide band gap semiconductor such as silicon carbide (SiC).

Supercritical fluid extraction (SFE) is an innovative green technology for the extraction of phytochemicals from plants. Therefore, this study aimed to evaluate the application of SFE and to optimize the extraction condition of the Thai herbal formula, Kleeb Bua Daeng (KBD). A Box-Behnken design (BBD) with response surface methodology was used to determine the effect of extraction time, temperature, and pressure on response variables including extraction yield, total phenolic content (TPC), total flavonoid content (TFC), total carotenoid content (TCC) and total anthocyanin content (TAC) of KBD formula. The highest percentage extraction yield, TPC, TFC, and TCC of the extracts were 3.81%, 464.56 mg gallic acid equivalents/g extract, 217.19 mg quercetin equivalents/g extract, and 22.26 mg β-carotene equivalents/g extract, respectively. The results indicated that SFE is a suitable method of extraction for green recovery of some phytochemicals from the KBD formula.

Scator set, introduced by Fernández-Guasti and Zaldívar, is endowed with a very peculiar non-distributive product. In this paper we consider the scator space of dimension $1+2$ and the so called fundamental embedding which maps the subset of scators with non-zero scalar component into 4-dimensional space endowed with a natural distributive product. The original definition of the scator product is induced in a straightforward way. Moreove, we propose an extension of the scator product on the whole scator space, including scators with vanishing scalar component.

Bronchopulmonary dysplasia (BPD) is the most common complication after preterm birth. Pasteurized donor human milk (DHM) has increasingly become the standard of care for very preterm infants over the use of preterm formula (PF) if mother’s own milk (MOM) is unavailable. Studies have reported beneficial effects of DHM on BPD. We conducted a systematic review and meta-analysis of randomized controlled trials (RCTs) and observational studies on the effects of DHM on BPD and other respiratory outcomes. Eighteen studies met the inclusion criteria. Meta-analysis of RCT’s could not demonstrate that supplementation of MOM with DHM reduced BPD when compared to PF (3 studies, risk ratio [RR] 0.89, 95% confidence interval [CI] 0.60–1.32). However, meta-analysis of observational studies showed that DHM supplementation reduced BPD (8 studies, RR 0.78, 95% CI 0.67–0.90). An exclusive human milk diet reduced the risk of BPD, compared to a diet with PF and/or bovine milk-based fortifier (3 studies, RR 0.80, 95% CI 0.68–0.95). Feeding raw MOM, compared to feeding pasteurized MOM, protected against BPD (2 studies, RR 0.77, 95% CI 0.62–0.96). In conclusion, our data suggest that DHM protects against BPD in very preterm infants, but pasteurization of human milk reduces the benefit.

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.

Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of the solution, without calculating the full inversion operator and the complete solution. It is a well-known fact about partial differential equations that the solution at each discretisation point depends on the solutions at all other discretisation points. Therefore, it is impossible to compute the solution only at one point, without calculating the solution at all other points. The standard numerical methods like a conjugate gradient or Gauss elimination compute the whole solution and/or the complete inverse operator. We suggest a method which can compute the solution of the given partial differential equation 1) at a point; 2) at few points; 3) on an interface; or a functional of the solution, without computing the solution at all points. The required storage cost and computational resources will be lower as in the standard approach. With this new method, we can speed up, for instance, computation of the innovation in filtering or the likelihood distribution, which measures the data misfit (mismatch). Further, we can speed up the solution of the regression, Bayesian inversion, data assimilation, and Kalman filter update problems. Applying additionally the hierarchical matrix approximation, we reduce the cubic computational cost to almost linear $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the hierarchical matrix approximation error, the computed solution is exact. One of the disadvantages of this method is the need to modify the existing deterministic solver.

Background: this retrospective study aimed to analyze the results of the combination of the Haigis formula and Total Keratometry (TK) to calculate the IOL power in eyes with previous corneal refractive surgery. Methods: The TK value provided by the IOL Master 700 (Carl Zeiss Meditec) was introduced into the Haigis formula; the mean prediction error (PE), mean absolute error (MAE), median absolute error (MedAE) and percentage of eyes with a PE within ±0.25 D, ±0.5 D, ±0.75 D and ±1.00 D were calculated. Results: Ninety-three eyes of 93 patients with previous laser refractive surgery were evaluated. Two groups were defined: Group A included 51 previously myopic eyes, Group B included 42 previously hyperopic eyes. The mean PE in Group A was +0.09 ±0.44 D and 76.47% of eyes had a PE within ±0.50 D. In Group B the mean PE was -0.15 ±0.46 D and 66.67% of eyes had a PE within ±0.50 D. Discussion: The Haigis formula combined with TK is one of the most accurate available methods for IOL power calculation in eyes with prior myopic and hyperopic corneal refractive surgery. In such eyes the results are similar to or better than those reported in previous studies.

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

The polynomial functional specification is widely used in environmental and ecological economics. When different powers of the same variable enter to the specification the elasticity of explained variable with respect to that variable should be obtained using the elasticity formula. Since, in this case the elasticity itself is a function, one need to calculate it at a certain point, to have a general idea about the response of explained variable to the change in explanatory variable. The mainly used point is mean of the variables entering to the elasticity formula. Sometimes, the minimum and maximum points also used for comparison purposes. One should careful in interpreting the response of explanatory variable to the change in the variable entering to the specification with its different powers. This study revisits some methodological points regarding the calculation and interpretation of responses in the above-mentioned cases.

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

Schrödinger dynamics is a nonlocal process. Not only does local perturbation affect instantaneously the entire space, but the effect decays slowly. When the wavefunction is spectrally bounded, the Schrödinger equation can be written as a universal set of ordinary differential equations, with universal coupling between them, which is related to Euler’s formula. Since every variable represents a different local value of the wave equation, the coupling represents the dynamics’ nonlocality. It is shown that the nonlocal coefficient is inversely proportional to the distance between the centers of these local areas. As far as we know, this is the first time that this inverse square law was formulated.

Introduction: This empirical study investigates the impact of two distinct mathematical problem-solving methods – the Algebraic Formula Method and the Newton Sum Method – on enhancing ChatGPT's competence in effectively solving quadratic root functions. The integration of Artificial Intelligence (AI) into mathematical problem-solving has paved the way for innovative approaches. In this study, we delve into the Algebraic Formula Method and the Newton Sum Method, essential techniques for solving quadratic root functions. We aim to showcase the profound influence of these methods on ChatGPT's capacity to excel in solving quadratic equations. Evidence Through concrete evidence, we demonstrate ChatGPT's adept utilization of the Newton Sum Method for quadratic root function calculations. While ChatGPT can compute quadratic root functions of the form α^15 + β^15 using this method, its proficiency in using algebraic formula methods typically extends only up to α^4 + β^4. This marked discrepancy underscores the pivotal role that different methods play in amplifying the AI system's mathematical capabilities Result The results of this study provide concrete evidence of ChatGPT's superior utilization of the Newton Sum Method for calculating quadratic root functions. The model adeptly computes expressions of the form α^15 + β^15 using this method, while its proficiency using algebraic formula methods is generally limited to α^4 + β^4. This striking discrepancy underscores the transformative impact that different methods can have on elevating the AI system's mathematical prowess. Conclusion :Pushing Boundaries: Pioneering Novel Maths Approaches for Overcoming Limitations in AIThis study serves as an illuminating testament to the significance of pioneering innovative methodologies, rules, theorems, or formulas to surmount the current limitations in AI systems like ChatGPT. These innovative pursuits hold the key to unlocking the untapped potential that lies within, propelling AI systems to greater heights of proficiency. In essence, they offer a strategic pathway towards expanding the capabilities of AI and pushing the boundaries of what can be achieved. Discussion The outcomes derived from this study underscore the significant influence wielded by the method selection in augmenting the mathematical competencies of ChatGPT. Particularly noteworthy is the application of the Newton Sum Method, which surfaces as a compelling exemplar. This method serves as a pivotal conduit through which the model surpasses its prior constraints, allowing it to venture into the realm of calculations entailing higher exponents. Implications and Future Research: These findings not only contribute to AI's mathematical competencies but also emphasize the need for pioneering new methods, rules, theorems, or formulas to further enhance AI systems like ChatGPT. Future research could explore the development of novel mathematical techniques tailored to AI systems, thus expanding their capabilities across diverse problem-solving domains.

Weeds are a permanent component of anthropogenic ecosystems, which requires strict control to avoid the accumulation of their long–lasting seeds in the soil. With high crop infestation, many elements of crop production technologies (fertilization, productive varieties, growth stimulators, etc.) turn out to be practically meaningless due to high yield losses. Intensive use of chemical herbicides (CH) has led to undesirable consequences: contamination of soil and wastewater, accumulation of their residues in the crop, the emergence of CH-resistant populations of weeds. In this regard, the development of environmentally friendly CH with new mechanisms of action is relevant. Natural phytotoxins of plant or microbial origin may be directly explored in herbicidal formulations (biorational CH) or indirectly as scaffolds for nature-derived CH. This review considers: 1) the main current trends in the development of CH that may be important for enhancement of biorational herbicides; 2) advances in the development and practical application of natural compounds for weed control; 3) the use of phytotoxins as prototypes of synthetic herbicides. Some modern approaches such as computational methods of virtual screening and design herbicidal molecules, development of modern formulations, determination of molecular targets are stressed as important to make exploration of natural compounds more effective.

As shown previously, a relation between the superconducting transition temperature and some characteristic distance in the crystal lattice holds, which enables the calculation of the superconducting transition temperature, Tc, based only on the knowledge of the electronic configuration and of some details of the crystallographic structure. This relation was found to apply for a large number of superconductors, including the high-temperature superconductors, the iron-based materials, alkali fullerides, metallic alloys, and element superconductors. When applying this scheme called Roeser-Huber formula to Moiré-type superconductivity, i.e., magic-angle twisted bi-layer graphene (tBLG) and bi-layer WSe2, we find that the calculated transition temperatures for tBLG are always higher than the available experimental data, e.g., for the magic angle 1.1∘, we find Tc≈ 4.2–6.7 K. Now, the question arises why the calculation produces larger Tc’s. Two possible scenarios may answer this question: (1) The given problem for experimentalists is the fact that for electric measurements always substrates/caps are required to arrange the electric contacts. When now discussing superconductivity in atomically thin objects, also these layers may play a role forming the Moiré patterns. The consequence of such substrate-induced super-Moiré patterns is that the resulting Moiré pattern always will show a larger cell size, and thus, a lower Tc of the final structure will result. (2) A correction factor to the Roeser-Huber formalism may be required to account for the low charge carrier density of the tBLG. Here, we test both scenarios and find that the introduction of a correction factor η enables a proper calculation of Tc, reproducing the experimental data. We find that η depends exponentially on the value of Tc.

As shown previously, a relation between the superconducting transition temperature and some characteristic distance in the crystal lattice holds, which enables the calculation of the superconducting transition temperature, Tc, based only on the knowledge of the electronic configuration and of some details of the crystallographic structure. This relation was found to apply for a large number of superconductors, including the high-temperature superconductors, the iron-based materials, alkali fullerides, metallic alloys, and element superconductors. When applying this scheme called Roeser-Huber formula to Moiré-type superconductivity, i.e., magic-angle twisted bi-layer graphene (tBLG) and bi-layer WSe2, we find that the calculated transition temperatures for tBLG are always higher than the available experimental data, e.g., for the magic angle 1.1∘, we find Tc≈ 4.2–6.7 K. Now, the question arises why the calculation produces larger Tc’s. Two possible scenarios may answer this question: (1) The given problem for experimentalists is the fact that for electric measurements always substrates/caps are required to arrange the electric contacts. When now discussing superconductivity in atomically thin objects, also these layers may play a role forming the Moiré patterns. The consequence of such substrate-induced super-Moiré patterns is that the resulting Moiré pattern always will show a larger cell size, and thus, a lower Tc of the final structure will result. (2) A correction factor to the Roeser-Huber formalism may be required to account for the low charge carrier density of the tBLG. Here, we test both scenarios and find that the introduction of a correction factor η enables a proper calculation of Tc, reproducing the experimental data. We find that η depends exponentially on the value of Tc.

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

In this paper, we investigate a new subclass of univalent functions defined by a generalized differential operator, and obtain some interesting properties of functions belonging to the class R^m λµ ( α, β, γ, υ ).

In this paper we survey some recent results due to the author concerning various inequalities and approximations for the finite Hilbert transform of a function belonging to several classes of functions, such as: Lipschitzian, monotonic, convex or with the derivative of bounded variation or absolutely continuous. More accurate estimates in the case that the higher order derivatives are absolutely continuous, are also provided. Some quadrature rules with error bounds are derived. They can be used in the numerical integration of the finite Hilbert transform and, due to the explicit form of the error bounds, enable the user to predict a priory the accuracy.

This paper presents the symmetries of differential equations associated with one-variable and Bivariate Hermite polynomials by proposing a representation of Lie algebra for these differential operators. Applying the Baker-Campbell-Hausdorff formula to these algebras, results in new relations and generating functions in one-variable and Bivariate Hermite polynomials. A general form of representation for other orthogonal polynomials such as Laguerre polynomials is introduced.

In the energy-momentum density expressions for a relativistic perfect fluid with a bulk motion, one comes across a couple of pressure-dependent terms, which though well known, are to an extent, lacking in their conceptual basis and the ensuing physical interpretation. In the expression for the energy density, the rest mass density along with the kinetic energy density of the fluid constituents due to their random motion, which contributes to the pressure as well, are already included. However, in a fluid with a bulk motion, there are, in addition, a couple of explicit, pressure-dependent terms in the energy-momentum densities, whose presence to an extent, is shrouded in mystery, especially from a physical perspective. We show here that one such pressure-dependent term appearing in the energy density, represents the work done by the fluid pressure against the Lorentz contraction during transition from the rest frame of the fluid to another frame in which the fluid has a bulk motion. This applies equally to the electromagnetic energy density of electrically charged systems in motion and explains in a natural manner an apparently paradoxical result that the field energy of a charged capacitor system decreases with an increase in the system velocity. The momentum density includes another pressure-dependent term, that represents an energy flow across the system, due to the opposite signs of work being done by pressure on two opposite sides of the moving fluid. From Maxwell's stress tensor we demonstrate that in the expression for electromagnetic momentum of an electric charged particle, it is the presence of a similar pressure term, arising from electrical self-repulsion forces in the charged sphere, that yields a natural explanation for the famous, more than a century old, 4/3 factor in the electromagnetic mass.

Species richness is a widely used measure for assessing the diversity of a particular area. However, observed richness often underestimates the true richness due to resource limitations, particularly in the small-sized sample or highly heterogeneous assemblage. Estimating species richness in a large-scale region typically involves an integrated data set consisting of subsamples collected independently from different subregions. However, the pooled sample of integrated data is no longer a random sample from the entire region, and the use of different sampling schemes results in variations in data formats. Consequently, employing a single sampling distribution to model the pooled sample becomes impractical, rendering existing richness estimators inadequate. This study theoretically explains the applicability of Chao's lower bound estimators in estimating species richness for large-scale areas using the pooled sample. Additionally, a new nonparametric estimator is introduced, which adjusts the bias of Chao's lower bound estimator by leveraging the Good-Turing frequency formula. This proposed estimator only utilizes the pooled sample's singleton, doubleton, and tripleton richness. Simulated data sets across various models are employed to demonstrate the statistical performance of the estimator, showcasing its ability to reduce bias and provide accurate 95% confidence intervals. Real data sets are also utilized to illustrate the practical application of the proposed approach.

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n^2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials D_m(n,k) and U_m(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n^2m+1.

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDE) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDE) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function and they are approximated respectively by using two multilayer neural networks as function approximators. Thus, deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and high-dimensional option pricing model with default risk. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

This paper is a continuation of previous papers on unification theories publishes in AXIOMS. We present results about the (modified) Yang-Baxter equation, Euler’s formula, dual numbers, coalgebra structures, non-associative structures, differential geometry, and (mean) inequalities. We will also attempt to relate our discussion to some Brain Studies and Machine Learnig.

In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.

We consider several unification problems in mathematics. We refer to transcendental numbers. Also, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

Introduction The authors reviewed the published evidence on the presence of oligosaccharides in human milk (HMO) and their benefits in in vitro and in vivo studies, as well as to evaluate published data on the addition of some of HMOs to infant formula. Method PubMed was searched from inception to April 2018 Results The presence of HMOs is a dynamic process, determined by many factors such as duration of lactation, environmental and genetic factors. HMOs may support immune function development and provide protection against infectious diseases directly through the interaction of the gut epithelial cells or indirectly through the modulation of the gut microbiota, including the stimulation of the bifidobacteria. Only a few of the many HMOs can be made industrially, and are added to infant formula. They have been shown to be safe and well tolerated, including normal growth and a trend towards health benefit. Conclusions HMOs are one of the major differences between cow’s milk and human milk , and evidence indicates that these components do have a health promoting benefit. The addition of one or two of these components to infant formula is safe, and brings infant formula closer to human milk.

In the upper charged layers of the atmosphere, the plasma is very rarefied. The collisions between its molecules are almost non-existent, and the driving forces behind them are the Lorentz forces resulting from the electric and magnetic fields. For this reason, we are interested in studying the behavior of non-collision plasmas because of its essential applications, such as the movement of satellites in the charged atmosphere. In this paper, the flow problem of collisionless gaseous plasma is examined. For that propose, we solve the unsteady Vlasov-Maxwell system of non-linear partial differential equations analytically. Methods of moments and traveling wave parameters are used to acquire an exact solution. Specific macroscopic properties of collisionless gaseous plasma are calculated along with electrical and magnetic fields. Further, thermodynamic estimation, such as entropy and entropy production, is presented. Those calculations allow us to measure the consistency with the laws of non-equilibrium thermodynamics. Relations between internal energy modification participations are predicted using Gibbs' equation for collisionless plasma. The modification effect of internal energies due to electro-magnetic fields is found to be small compared with the internal energy change due to the effect of entropy. That is because these fields are self-induced by plasma particles due to the sudden movement of the rigid plane plate. The results are accomplished according to the typical argon gaseous plasma model. Three-dimensional diagrams showing the measured variables are drawn to investigate and discuss their behavior. The problem has many commercial applications for the movement of objects in the charged atmosphere.

Human milk composition is dynamic and substitute formulae are intended to mimic its protein content. The purpose of this study was to investigate the potentiality of matrix-assisted laser desorption/ionization-time-of-flight mass spectrometry (MALDI-TOF MS) followed by multivariate data analyses as a tool to analyze peptide profiling of mammalian, human and formula milks. Breast milk samples from women at different lactation stages (2 (n = 5), 30 (n = 6), 60 (n = 5), and 90 (n = 4) days postpartum), and milk from donkeys (n = 4), cows (n = 4), buffaloes (n = 7), goats (n = 4), ewes (n = 5), and camels (n = 2) were collected. Different brands (n = 4) of infant formulae were also analyzed. Protein content (<30 kDa) was analyzed by MS and data were exported for statistical elaborations. Mass spectra for each milk closely clustered together, whereas different milk samples resulted well separated. Human samples formed a cluster in which colostrum constituted a well-defined subcluster. None of the milk formulae correlated with animal or human milk, although specifically characterized and well correlated each other. These findings propose MALDI-TOF MS milk profiling as an analytical tool to discriminate, in a blinded way, different milk types. As each formula has a distinct specificity, shifting a baby from one to another formula implies a specific proteomic exposure. These profiles may assist in milk proteomics for easiness of use and low cost consuming, suggesting that the MALDI-TOF MS pipelines may result useful for milk adulteration assessment but also for the characterization of banked milk specimens in paediatric clinical settings.

The core of risk aggregation in the Solvency II Standard Formula is the so-called square root formula. We argue that it should be seen as a means for the aggregation of different risks to an overall risk rather than being associated with variance-covariance based risk analysis. Considering the Solvency II Standard Formula from the viewpoint of linear geometry, we immediately find that it defines a norm and therefore provides a homogeneous and sub-additive tool for risk aggregation. Hence Euler's Principle for the reallocation of risk capital applies and yields explicit formulas for capital allocation in the framework given by the Solvency II Standard Formula. This gives rise to the definition of diversification functions, which we define as monotone, subadditive, and homogeneous functions on a convex cone. Diversification functions constitute a class of models for the study of the aggregation of risk, and diversification. The aggregation of risk measures using a diversification function preserves the respective properties of these risk measures. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Each L^p norm has this property, and any scalar product given by a non-negative positive semidefinite matrix does as well. In particular, the Standard Formula is a diversification function and hence a risk measure that preserves homogeneity, subadditivity, and convexity.

Reverse-order laws for generalized inverses of matrix products is a classic object of study in the theory of generalized inverses. One of the well-known reverse-order laws for a matrix product AB is (AB)^(i,...,j) = B^(i,...,j)A^(i,...,j), where (·)^i,...,j denotes an {i,...,j}-generalized inverse of matrix. Because {i,...,j}-generalized inverse of a general matrix is not necessarily unique, the relationships between both sides of the reverse-order law can be divided into four situations for consideration. In this article, we first introduce a linear mixed model y = ABβ + Aγ + ε, present two least-squares methodologies to estimate the fixed parameter vector  in the model, and describe the connections between the two least-squares estimators and the reverse-order laws for generalized inverses of the matrix product AB. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B^(i,...,j)A^(i,...,j), as well as necessary and sufficient conditions for B^(i,...,j)A^(i,...,j) to be invariant with respect to the choice of A^(i,...,j) and B^(i,...,j). We then present a unied approach to the 512 set inclusion problems {(AB)^(i,...,j) ⊇ {B^(i,...,j)A^(i,...,j)}for the eight commonly-used types of generalized inverses of A, B, and AB using the block matrix representation method (BMRM), matrix equation method (MEM), and matrix rank method (MRM), where {(·)^(i,...,j)} denotes the collection of all {i,...,j}-generalized inverse of a matrix.

In cricket, particularly the twenty20 format is most watched and loved by the people, where no one can guess who will win the match until the last ball of the last over. In India, The Indian Premier League (IPL) started in 2008 and now it is the most popular T20 league in the world. So we decided to develop a machine learning model for predicting the outcome of its matches. Winning in a Cricket Match depends on many key factors like a home ground advantage, past performances on that ground, records at the same venue, the overall experience of the players, record with a particular opposition, and the overall current form of the team and also the individual player. This paper briefs about the key factors that affect the result of the cricket match and the regression model that best fits this data and gives the best predictions. Cricket, the mainstream and widely played sport across India which has the most noteworthy fan base. Indian Premier League follows 20-20 format which is very unpredictable. IPL match predictor is a ML based prediction approach where the data sets and previous stats are trained in all dimensions covering all important factors such as: Toss, Home Ground, Captains, Favorite Players, Opposition Battle, Previous Stats etc, with each factor having different strength with the help of KNIME Tool and with the added intelligence of Naive Bayes network and Eulers strength calculation formula.

SARS-CoV-2 belongs to the group of RNA viruses with a pronounced tendency to mutate. Omicron BA.2.75 is a subvariant believed to be able to suppress the currently dominant BA.5. Omicron BA.2.75 is characterized by a greater infectivity compared to earlier Omicron variants. However, Gibbs energy of biosynthesis of virus particles is slightly less negative compared to those of other variants. Thus, the multiplication rate of Omicron BA.2.75 is lower than that of other variants. This leads to slower accumulation of newly formed virions and less damage to host cells, indicating evolution of SARS-CoV-2 towards decreasing pathogenicity.

The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. For example, the duality issue for multi-criteria and vector optimization problems could be solved using the complete-lattice approach, compare [11]. So far, it has been applied to set-valued dynamic risk measures (in the stochastic case), as discussed in Feinstein, Rudloff etc. (see [11], for example), but it has not been applied to deterministic calculus of variations and optimal control problems. In this paper, the following problem of set-valued optimization is considered: minimize the functional $$ \overline J_t[y]=\int_0^t \overline L(s,y(s),\dot y(s))\ ds + U_0(y(0)) $$ over all admissible arcs $y$, where $\overline L$ is the associated multifunction to a vector-valued Lagrangian $L$, the integral is in the Aumann sense and $U_0$ is the initial cost. A new concept of \emph{value function}, for which a Bellman's optimality principle holds, is introduced. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time $t$ and a directional derivative with respect to $x$ of the value function are defined, based on ideas close to the concepts in [12]. The value function is proved to be solution of a suitable Hamilton-Jacobi equation.

In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity between $m$-order polynomials in $T$ $$\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.$$

Abstract: Supplementation of infant formula with ingredients potentially able to manage, in a way positive for the host, the gut microbiota of infants, is nowadays widely used by infant food producers. The impact of this class of compounds, named prebiotics, on gut microbiota composition, was initially measured on the enrichment of Bifidobacterium and Lactobacillus spp., but the use of culture-independent analytical techniques has allowed to establish that a larger number of intestinal microorganisms are affected by the ingestion of prebiotics. However, despite a relevant number of scientific publications, actually there is no consensus on the amount of prebiotics to be administered daily, the potentially different actions exerted by prebiotics which differ in biochemical structures, as well as the impact of different percentages of the same prebiotics when used in mixtures and/or combination. This paper is aimed to critically review the available data on the use of prebiotics in infant formula, with special attention paid to identify a link between, the dosage used, the mixture composition and the observed outcomes, keeping in mind the influences in nutrition quality and growth influence.

During flux weakening operation time (FWOT), the total iron loss rises and affects the accuracy of loss prediction and efficiency especially if a large range of FWOT exists. Iron loss prediction is widely employed in investigations for a fast electrical machine analysis using 2-D FEA. This paper studies harmonic loss analytically by a steady-state equivalent circuit. Consideration of skin effects and iron saturation are utilized in order to examine the accuracy through the relative error distribution in the frequency domain of each model from 50 to 700 Hz. Additionally, this comparative study presents a torque-frequency-flux density calculation over each single term of the modified IEM-formula. The analytical calculation is performed using 2-D FEA for a classic and modified IEM-formula along with experimental verifications on a surface-mounted PMSG for a wind generation application.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

This research aims to study the ultimate limit state (ULS) behaviour of stiffened panel under longitudinal compression by non-linear finite element method (NLFEM). There are different types of stiffeners being used in shipbuilding i.e. T-bar, flat-bar and angle-bar. However, this research focuses on the ultimate compressive strength behaviour of flat-bar stiffened panel. A total of 420 of reliable scenarios of flat-bar stiffened panel are selected for numerical simulation by ANSYS NLFEM. The ultimate strength behaviours obtained were used as data for the development of closed form shape empirical formulation. Recently, Kim et al. [1] proposed for advanced empirical formulation for T-bar stiffened panel and the applicability of the proposed formulation to flat-bar stiffened panel will be confirmed by this study. The accuracy of the empirical formulation obtained for flat-bar stiffened panel has been validated by FE simulation results of statistical analysis (R2 = 0.9435). The outcome obtained will be useful for ship structural designers in predicting the ultimate strength performance of flat-bar type stiffened panel under longitudinal compression.

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

This work presents an approach for detecting mathematical formulas in scanned document images. The proposed approach is end-to-end trainable. Since many OCR engines cannot reliably work with the formulas, it is essential to isolate them to obtain the clean text for information extraction from the document. Our proposed pipeline comprises a hybrid task cascade network with deformable convolutions and a Resnext101 backbone. Both of these modifications help in better detection. We evaluate the proposed approaches on the ICDAR-2017 POD and Marmot datasets and achieve an overall accuracy of 96% for the ICDAR-2017 POD dataset. We achieve an overall reduction of error of 13%. Furthermore, the results on Marmot datasets are improved for the isolated and embedded formulas. We achieved an accuracy of 98.78% for the isolated formula and 90.21% overall accuracy for embedded formulas. Consequently, it results in an error reduction rate of 43% for isolated and 17.9% for embedded formulas.

Black hole collision produce gravitational radiation which is generally thought in a quantum limit to be gravitons. The stretched horizon of a black hole contains quantum information, or a form of quantum hair, which in a coalescence of black holes participates in the generation of gravitons. This may be facilitated with a Bohr-like approach to black hole (BH) quantum physics with quasi-normal mode (QNM) approach to BH quantum mechanics. Quantum gravity and quantum hair on event horizons is excited to higher energy in BH coalescence. The near horizon condition for two BHs right before collision is a deformed AdS spacetime. These excited states of BH quantum hair then relax with the production of gravitons. This is then argued to define RT entropy given by quantum hair on the horizons. These qubits of information from a BH coalescence should then appear in gravitational wave (GW) data. This is a form of the standard AdS/CF T correspondence and the Ryu-Takayanagi (RT) formula [1]. The foundations of physics is proposed to be quantum information and a duality between spacetime observables and quantum fields.

In times of water scarcity, water conservation measures from authorities are primarily directed to the residential sector—homeowners and renters who engage in individual activities to reduce water consumption. Business owners, manufacturers, and institutional workers—the ICI sector, generally, do not receive targeted information and education on water savings measures, and when they do, the information, typically, refers to indoor technologies, and behavior choice. In answer to this shortfall, this research illustrates how the ICI sector might engage in outdoor water conservation and, thus, realize water savings using the well-established Pittenger’s Simplified Landscape Irrigation Design Estimation (SLIDE) formula that estimates outdoor water conservation/savings for the lesser studied industrial, commercial, and institutional (ICI) sector. We applied the SLIDE formula five diverse climate areas in Texas to demonstrate the potential water savings in each, as well as identifying the type of water technology measures that yielded the greatest water savings for outdoor ICI landscapes. Overall, the greatest water savings for outdoor ICI landscapes was realized through smartscape design with soil moisture sensors-SMS technology, second best. Our findings will provide ICI facility owners/managers with knowledge and examples of a simplified system for quantitative decision making when considering choices among technologies and/or practices toward outdoor water conservation for their own facilities.

3D Zernike moments based on 3D Zernike polynomials have been successfully applied to the field of voxelized 3D shape retrieval and have attracted more attention in biomedical image processing. As the order of 3D Zernike moments increases, both computational efficiency and numerical accuracy decrease. Due to this phenomenon, a more efficient and stable method for computing high-order 3D Zernike moments was proposed in this study. The proposed recursive formula for computing 3D Zernike radial polynomials combines the recursive calculation of spherical harmonics to develop a voxel-based algorithm for the calculation of 3D Zernike moments. The algorithm was applied to the 3D shape Michelangelo's David with a size of 150×150×150 voxels. As compared to the method without additional acceleration, the proposed method uses a group action of order sixteen orthogonal group and saving unnecessary iterations, the factor of speed-up is 56.783±3.999 when the order of Zernike moments is between 10 and 450. The proposed method also obtained an accurate reconstructed shape with the error rate (normalized mean square error) of 0.00 (4.17×10^-3) when the reconstruction was computed for all moments up to order 450.

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

We point out an inconsistency in Newton's equations of celestial mechanics. A set of differential equations implied by Newton's equations are shown to be free of this inconsistency. We then investigate the integrals of motion associated with this relative difference system.

This paper aims to present the basic assumptions for creation of social Fröhlich condensate and attract attention of other researchers (both from physics and socio-political science) to the problem of modelling of stability and order preservation in highly energetic society coupled with social energy bath of high temperature. The model of social Fröhlich condensation and its analysis are based on the mathematical formalism of quantum thermodynamics and field theory (applied outside of physics). The presented quantum-like model provides the consistent operational model of such complex socio-political phenomenon as Fröhlich condensation. The model of social Fröhlich condensation is heavily based on theory of open quantum systems. Its consistent elaboration needs additional efforts. Evidence of such phenomenon as social Fröhlich condensation is demonstrated by stability of modern informationally open societies Approaching the state of Fröhlich condensation is the powerful source of social stability. Understanding its informational structure and origin may help to stabilize the modern society. Application of the quantum-like model of Frhlich condensation in social and political sciences is really the novel and original approach to mathematical modeling of social stability in society exposed to powerful information radiation from mass-media and internet based sources.

The aim of this article is to establish a new class of unified integrals associated with the generalized Lommel-Wright functions, which are expressed in terms of Wright Hypergeometric function.Some integrals involving trigonometric,generalized Bessel function and Struve functions are also indicated as special cases of our main results.further, with the help of our main results and their special cases, we obtain two reduction formulas for the Wright hypergeometric function.

The human ankle is a complex joint, most commonly represented as talocrural and subtalar axes. It is difficult to locate and take in vivo measurements of the ankle joint. There are no instruments for patients lying on a bed or the floor; that can be used in outdoor or remote sites. We have developed a "Turmell-meter" to address these issues. We started with the study of ankle anatomy and anthropometry, then we used the product of exponentials’ formula to visualize the movements. Furthermore, we built a prototype using human proportions and statistics. For pose estimation, we used a trilateration method by applying tetrahedral geometry. Additionally, we computed the axis direction by fitting 3D circles, plotting the manifold and chart as an ankle joint model. We presented the results of simulations, a prototype comprising 45 parts, specifically designed draw-wire sensors, and electronics. Finally, we tested the device by capturing positions and fitting them into the bi-axial ankle model as a Riemannian manifold. The Turmell-meter is intended to be a hardware platform for human ankle joint axis estimation, it is adjustable and has an easy setup. The proposed model has the properties of a chart in a geometric manifold, we provided the details

Nowadays, brain signal processing is performed rapidly in various brain-computer interface (BCI) applications. Most researchers focus on developing new methods for the future or improving the basic implemented models to identify the optimum standalone feature set. Our research focuses on four ideas. One of them introduces future communication models, and the others are for improving old models or methods. These are: 1) new communication imagery model instead of speech imager using the mental task: Due to speech imagery is very difficult, and it is impossible to imagine sound for all of the characters in all of the languages. Our research introduces a new mental task model for all languages that call Lip-sync imagery. This model can use for all characters in all languages. This paper implemented two lip-sync for two sounds, characters or letters. 2) New combination Signals: Selecting an inopportune frequency domain can lead to inefficient feature extraction. Therefore, domain selection is so important for processing. This combination of limited frequency ranges proposes a preliminary for creating Fragmentary Continuous frequency. For the first model, two s intervals of 4 Hz as filter banks were examined and tested. The primary purpose is to identify the combination of filter banks with 4Hz (scale of each filter bank) from the 4Hz to 40Hz frequency domain as new combination signals (8Hz) to obtain well and efficient features using increasing distinctive patterns and decreasing similar patterns of brain activities.3) new supplement bond graph classifier for SVM classifier: When SVM linear uses in very noisy, the performance is decreased. But we introduce a new bond graph linear classifier to supplement SVM linear in noisy data. 4) a deep formula recognition model: it converts the data of the first layer into a formula model (formula extraction model). The main goal is to reduce the noise in the subsequent layers for the coefficients of the formulas. The output of the last layer is the coefficients selected by different functions in different layers. Finally, the classifier extracts the root interval of the formulas, and the diagnosis does based on the root interval. For all of the ideas achieved the results of implementing methods. The results are between 55% to 98%. Less result is 55% for the deep detection formula, and the highest result is 98% for new combination signals.