Addition Formula Research Articles

A Levi–Civita Equation on Monoids, Two Ways

Abstract We consider the Levi–Civita equation f ( x y ) = g 1 ( x ) h 1 ( y ) + g 2 ( x ) h 2 ( y ) f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g 1, g 2, h 1, h 2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

Predicting compressive strength and behavior of ice and analyzing feature importance with explainable machine learning models

Building and using ice-related models is challenging due to the complexity of the material. A common issue, shared by both material models and (semi-)empirical approaches, is estimating unknown input parameters such as compressive strength. This is often done with additional empirical formulas which have drawbacks, e.g., they are based on a limited amount of data. Regarding material modeling, a strongly related problem is the prioritization of effects to include in the model. This is mostly done based on a subjective mix of knowledge, model purpose, and experimental studies limited to that purpose, which risks overlooking effects or interaction of effects, and limits transferability of material models to other scenarios.To tackle these issues, a hybrid approach of domain knowledge and explainable machine learning was used. A large ice test database was compiled to train machine learning models to predict compressive strength and behavior type. The machine learning models’ predictions were more accurate than existing empirical or analytical approaches and can thus be used as an alternative, though less straightforward, tool for such predictions. Further, the SHAP explainable AI method was applied to the predictions. Impact rankings of experimental parameters and interaction effects between parameters were analyzed and discussed in terms of ice mechanics. Top features were strain rate, triaxial stress state, temperature, and loading direction, but impact rankings were highly dependent on prediction target and type of ice. Few interaction effects were found. The approach adds objectivity to the prioritization of effects for material modeling and generated further insights into ice mechanics. It is also considered useful for other natural materials or generally when there is more data than knowledge.

Influences of Impurity Incorporation in Electroplated Cu on the SnAgCu and Ni-Containing SnAgCu Solder Joints

SnAgCu and Ni-containing SnAgCu alloys are Pb-free solders widely used to join with Cu to construct the solder joints. Electrodeposition is a technology commonly used to fabricate Cu but co-deposition of organic impurities originating from additives is an inevitable reliability issue. This study investigates the impurity effect on the voiding propensity in the two solder joints (SnAgCu/Cu and SnAgCu-Ni/Cu) subjected to thermal aging at 200 °C. Results show that a high level of impurity incorporation causes massive void propagation along the SnAgCu/Cu and SnAgCu-Ni/Cu interface. Reduction of the impurity concentration by precise control of the additive formulas can weaken the impurity effect and effectively suppress the void propagation. The weakening phenomenon of the impurity effect is more pronounced in the SnAgCu-Ni/Cu joint, indicating that suppression of the Cu3Sn growth as well as Kirkendall voids by Ni addition is also helpful in reducing the influences of impurities.

Ray Tracing versus Thin-Lens Formulas for IOL Power Calculation Using Swept-Source Optical Coherence Tomography Biometry

PurposeTo evaluate the ray tracing method's accuracy employing Okulix ray tracing software and thin-lens formulas to calculate intraocular lens (IOL) power using a swept-source optical coherence tomography (SS-OCT) biometer (OA2000).MethodsA total of 188 eyes from 180 patients were included in this study. An OA-2000 optical biometer was used to collect biometric data. The predicted postoperative refraction based on thin-lens formulas including SRK/T, Hoffer Q, Holladay 1, and Haigis formulas and the ray tracing method utilizing the OKULIX software was determined for each patient. To compare the accuracy of approaches, the prediction error and the absolute prediction error were determined.ResultsThe mean axial length (AL) was 23.66 mm (range: 19–35). In subgroup analysis based on AL, in all ranges of ALs the ray tracing method had the lowest mean absolute error (0.56), the lowest standard deviation (SD; 0.55), and the greatest proportion of patients within 1 diopter of predicted refraction (87.43%) and the lowest absolute prediction error compared to the other formulas (except to SRK/T) in the AL range between 22 and 24 mm (all P 0.05). In addition, the OKULIX and Haigis formulas had the least variance (variability) in the prediction error in different ranges of AL.ConclusionThe ray tracing method had the lowest mean absolute error, the lowest standard deviation, and the greatest proportion of patients within 1 diopter of predicted refraction. So, the OKULIX software in combination with SS-OCT biometry (OA2000) performed on par with the third-generation and Haigis formulas, notwithstanding the potential for increased accuracy in the normal range and more consistent results in different ranges of AL.

Design model for the compressive strength of angle members including welded end-joints

The practical design of single steel angle members in compression has to cover multiple effects. Due to the commonly eccentric connection on only one angle leg (bolted or welded), additional bending moments are acting on the member, leading to a complex load carrying behaviour with flexural and/or lateral torsional buckling phenomena. Furthermore, type and size of rotational restraints at the member’s ends (provided by the adjacent structure) significantly influence the compression member capacity of these members.Within this paper, a recently developed design approach for hot-rolled angle members in compression is introduced. The presented method allows for calculating the internal forces based on elastic second order theory for an individual member with eccentricities and rotational spring stiffness at both ends. This calculation can be carried out by means of conventional structural analysis software. Detailed analytical models for the estimation of appropriate spring stiffness values for several practical applications in buildings with two-bolt connections have recently been presented by the authors. The proposed design method is now enhanced by means of additional formulae for angle members welded to the adjacent structure. Detailed background information on the derivation of these newly developed spring stiffness formulae is given. Finally, the accuracy of the design model for this enlarged range of application for angle members with welded joints is shown through comparison with sophisticated finite element calculations, code provisions (AISC 360 and EN 1993-1-1) and experimental tests from literature.

Some families of closed classes in Pk defined by additive formulas

Abstract We analyse closed classes in k-valued logics containing all linear functions modulo k. The classes are determined by divisors d of a number k and canonical formulas for functions. We construct the lattice of all such classes for k = p 2, where p is a prime, and construct fragments of the lattice for other composite k.

Sodium Selenite Modulates IDO1/Kynurenine, TLR4, NF-κB and Bcl2/Bax Pathway and Mitigates Acetic Acid-Induced Colitis in Rat.

Colitis is a main presentation of inflammatory bowel disease (IBD) and yet, has no definitive cure. Currently, corticosteroids, anti-tumor necrosis factor (anti-TNF) agents and 5-aminosalicylic acid derivatives are prescribed for management of colitis. Except their failure rate, they are not always tolerated because of their severe adverse effects. Additive formulas with fewer adverse effects may improve the treatment of colitis. In this study, colitis was induced with intra-rectal injection of three concentrations of acetic acid (4, 6 and 8 v/v). Each group received sodium selenite (0.5 mg/kg) or saline, gavaged on days 0 and 1 for treatment. Two days after induction of colitis, rats were sacrificed and the end part of their colons were resected for macroscopic and microscopic evaluation and molecular measurement. Sodium selenite improved macroscopic and microscopic view of the colon, decreased cryptitis, crypt abscess and inflammatory cells infiltration and partly maintained mucosal structure. Sodium selenite markedly reduced tissue levels of malondialdehyde (MDA), TNF-α and interferon γ (INF-γ) and decreased myeloperoxidase (MPO) activity. Treatment with sodium selenite also significantly downregulated IL17, IL22, indoleamine 2,3-dioxygenase (IDO1), and kynurenine levels. Western blotting revealed that sodium selenite prevented apoptosis by increasing bcl2/Bax ratio. Furthermore, our findings showed that sodium selenite significantly downregulated the upstream inflammatory molecules such as nuclear factor kappa B (NF-κB) and toll-like receptor 4 (TLR4) in colitis. These findings show that sodium selenite alleviates inflammatory response and oxidative stress and protects against colitis.

A central limit theorem for integrals of random waves

Abstract We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.

Densely generated 2D q-Appell polynomials of Bessel type and q-addition formulas

The article aims to introduce a densely generated class of $2D$ $q$-Appell polynomials of Bessel type via generating equation and to establish its $q$-determinant form. It is advantageous to consider the $2D$ $q$-Bernoulli, $2D$ $q$-Roger Szeg\"{o} and $2D$ $q$-Al-Salam Carlitz polynomials of Bessel type as their special members. The $q$-determinant forms and certain $q$-addition formulas are derived for these polynomials. The article concludes with a brief view on discrete $q$-Bessel convolution of the $2D$ $q$-Appell polynomials.

High-degree Compression Functions on Alternative Models of Elliptic Curves and their Applications

This paper presents method for obtaining high-degree compression functions using natural symmetries in a given model of an elliptic curve. Such symmetries may be found using symmetry of involution [–1] and symmetry of translation morphism τT = P + T, where T is the n-torsion point which naturally belongs to the E(𝕂) for a given elliptic curve model. We will study alternative models of elliptic curves with points of order 2 and 4, and specifically Huff’s curves and the Hessian family of elliptic curves (like Hessian, twisted Hessian and generalized Hessian curves) with a point of order 3. We bring up some known compression functions on those models and present new ones as well. For (almost) every presented compression function, differential addition and point doubling formulas are shown. As in the case of high-degree compression functions manual investigation of differential addition and doubling formulas is very difficult, we came up with a Magma program which relies on the Gröbner basis. We prove that if for a model E of an elliptic curve exists an isomorphism φ : E → EM, where EM is the Montgomery curve and for any P ∈ E(𝕂) holds that φ(P) = (φx(P), φy(P)), then for a model E one may find compression function of degree 2. Moreover, one may find, defined for this compression function, differential addition and doubling formulas of the same efficiency as Montgomery’s. However, it seems that for the family of elliptic curves having a natural point of order 3, compression functions of the same efficiency do not exist.

Increasing the exclusive breastfeeding rate in a private hospital in UAE through quality improvement initiatives.

Mother's milk is the best, and ideal method for infant feeding. We found that this practice was not being followed in our hospital. A survey was conducted in the unit with regards to breastfeeding practices which revealed that most babies who were discharged from our nursery were on formula supplements. Our goal was to improve established breastfeeding rates in the unit by increasing the number of full-term healthy babies who were discharged on exclusive Breastfeeding. A project team was formed, and data were collected through direct observations and direct interviews with postnatal mothers. Exclusive Breastfeeding at discharge was defined as a baby being fully on breastfeed with no additional formula supplements for at least 12 hours prior to discharge. The primary outcome was to increase the percentage of babies being discharged on exclusive Breastfeeding. We used the FOCUS PDCA model to measure improvements and 8 PDCA cycles of 4 weeks duration were implemented to test the changes. The interventions we put in place led to a considerable nine times overall improvement in the established breastfeeding rates. Among all the interventions, the most promising results were observed during the PDCA cycles involving staff education, the introduction of antenatal classes for mothers, skin to skin contact and rooming in. Breastfeeding rates in the private sector with nursery services can be improved by reinforcing breastfeeding education for mothers in addition to training the maternal care staff, empowering them to promote and assist in breastfeeding.

Application of Forecasting as an Element of Effective Management in the Field of Improving Occupational Health and Safety in the Steel Industry in Poland

(1) Background: Every day, human beings fall victim to accidents. We implement solutions aimed at reducing accidents in everyday life, but we are not able to eliminate all accidents from our life. This article addresses the issue of forecasting accidents at work in the steel industry in Poland. Particular attention is paid to other accidents, given that those events are most often recorded in the sector under analysis. (2) Methods: The process of predicting quantitative data on the number of persons injured in other accidents in 2009–2018 employed Holt’s models: with an additive and multiplicative trend, with the trend smoothing effect in the multiplicative and additive formula. (3) Results: The forecasts prepared on the basis of Holt’s models and the combined model show a decreasing trend in the number of persons injured in other accidents in the steel sector, which is a positive development in the area of occupational safety and health. (4) Conclusions: The number of persons injured in other accidents at work in the steel sector shows a downward trend, which is significant and valid information for managers. The analysis of the results indicated that the combined forecast model best reflects the accidents at work in the steel industry.

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

Green Function of Generalized Time Fractional Diffusion Equation Using Addition Formula of Mittag-Leffler Function

In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.

Applications of a duality between generalized trigonometric and hyperbolic functions II

Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.

Dirichlet twists ofGLn-automorphicL-functions and hyper-Kloosterman Dirichlet series

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to $\operatorname{GL}_n$-automorphic $L$-functions.

Production and Supply Chain Operation Planning “Minuman Nanas” Innovation of Subang Honey Pineapple Beverage

Background – PT. Limatra runs the business of manufacturing and selling pineapple beverage products with various variants. PT Limatra provides packaged pineapple fruit juice products derived from 100% pineapple juice without the addition of preservatives. With 100% pineapple juice, of course it has a high vitamin C content as well. With the use of UHT (Ultra High Temperature) technology in the maklon industry and the use of aseptic paperpack packaging, it can produce good quality fruit juice products, maintain the freshness of the juice and have a relatively long product durability. PT Limatra provides solutions, namely: (1) Providing pineapple juice products with different flavors (mixed apple, mango and orange); (2) Providing pineapple juice products using aseptic paperpack packaging technology; (3) Using UHT technology to maintain the freshness and durability of the product; (4) Products that provide 100% pure fruit juice without additional formulas. To produce superior products requires an optimal operational plan. The method used is descriptive method. Results- From the results of the analysis and design of the operational plan, the Stages of Business/Business Establishment, Operational Goals and Targets, Operational Design, Operational Delivery, and Operational Cost Projections are generated. Keywords: Minuman Nanas, Subang Honey Pineapple Beverage, Production and Supply Chain Operation.

On $q$-hypergeometric Bernoulli polynomials and numbers

We introduce $q$-analogues of the hypergeometric Bernoulli polynomials in one and two real parameters and study several of their properties. Also we provide the inversion, the power representation, the multiplication and the addition formula for these polynomials. Classical results are recovered by limit transition.

Approximation and localized polynomial frame on conic domains

Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone.The main results provide construction of semi-discrete localized tight frame in weighted L2 norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a K-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the K-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain.