Terms Of Formula Research Articles

A New Approach on the Two-Dimensional Differential Transform

In recent years, new formulas of the two-dimensional differential transform have been proven by using the definition of the transform. In this work, we use a new approach based on the definition of the transform and the summation properties to prove the two-dimensional differential transform of the product of two functions, then we used this result to establish other useful formulas. This study shows that this procedure can be used to find formulas for many complicated terms. This enables us to apply the differential transform method on many types of partial differential equations. To demonstrate this approach, we applied the dimensional differential transform method on selected equations and compared our results with analytical solutions obtained by other methods

High resolution FTIR and diode laser spectroscopy of trifluoromethylacetylene and tetrafluoromethane in a supersonic jet expansion

We report high resolution infrared spectra in the CF stretching fundamental ranges of trifluoromethylacetylene (CFC≡C-H, 3,3,3-trifluoropropyne) and of tetrafluoromethane (CF, R14, PCF-14, Halocarbon14). For the experiments a diode laser setup combined with a supersonic jet expansion is described, achieving approximate rotational temperatures between 5 and 10 K for CF, using a pulsed slit nozzle providing effective optical path lengths of 3 to 4 cm in a single pass. Slightly higher rotational temperatures (near 30 K) for CFCCH greatly simplify the rotational-vibrational structure of the spectra for this molecule, for which we report also high resolution Fourier transform (FTIR) spectra at room temperature with nearly Doppler limited resolution (instrumental bandwidth 0.001 cm) using the Zürich prototype spectrometer ZP 2001 (Bruker IFS125). The analysis of the CF-symmetric stretching (A) fundamental provides accurate spectroscopic parameters of the ground and excited vibrational states with a band centre = (1253.4985 ± 0.0015) cm from a least squares adjustment to more than 1100 line data with a root mean square deviation less than 0.0003 cm. The low temperature spectra of the degenerate (F) fundamental of CF show a very simple structure of line clusters which can be analysed with an approximate term formula. The spectra are essentially instrument limited with a smallest effective line width near 0.002 cm. The assignment of the P(1) line at 1283.33693 cm in the supersonic jet spectra (about 10 K) provides a band centre in terms of the vibrational energy of the ( = 1, J = 0, F (F)) level at (1283.7193 ± 0.002) cm. Also the assignment of the isolated R(0) line at 1283.48420 cm provides the energy of the well defined ( = 1, J' = 1, A (A)) level. The results are discussed in relation to the data from the HITRAN tables, earlier FTIR data, and new frequency comb based data for CF. We also discuss possible studies of nuclear spin symmetry conservation in CF, for which lower temperatures would be required, and a concept to study parity violation in CF by a level crossing experiment using molecular beam spectroscopy.

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.

Use of standardised feeding protocols in neonatal units - A national survey of Australian neonatal nurses and dietitians

Background/Aim: Given the limited literature, this study aimed to describe current feeding practices as reported by neonatal staff, including use of Standardised Feeding Protocols (SFPs) and pasteurised donor human milk (PDHM). MethodsAn electronic survey was emailed to neonatal nurses and dietitians through professional organisations and networks. Descriptive and chi-square tests were conducted. Comments were summarised into themes. ResultsEligible respondents (n = 229) were female (98%), neonatal nurses (95%) with 17 (0.3–43) years of experience. Most (82%) observed SFPs in neonatal settings. Over 75% reported breast milk, term formula or low birth weight formula inclusion in SFPs. PDHM was included less often in SFPs (52%), particularly in regional/rural (10%) than capital areas (71%, p < 0.001). Patient weight and growth were common criteria used for commencing or ceasing fortification. Themes emerged supporting PDHM availability, and consistent SFPs across Australia. ConclusionAn opportunity to standardise evidence-based SFPs across neonatal settings exists.

Explicit closed algebraic formulas for Orlov–Scherbinn-point functions

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.

Penalised likelihood methods for phase-type dimension selection

Abstract Phase-type distributions are dense in the class of distributions on the positive real line, and their flexibility and closed-form formulas in terms of matrix calculus allow fitting models to data in various application areas. However, the parameters are in general non-identifiable, and hence the dimension of two similar models may be very different. This paper proposes a new method for selecting the dimension of phase-type distributions via penalisation of the likelihood function. The penalties are in terms of the Green matrix, from which it is possible to extract the contributions of each state to the overall mean. Since representations with higher dimensions are penalised, a parsimony effect is obtained. We perform a numerical study with randomly generated phase-type samples to illustrate the effectiveness of the proposed procedure, and also apply the technique to the absolute log-returns of EURO STOXX 50 and Bitcoin prices.

Learning Styles, Brain Dominance and Teaching Techniques: A Case Study Approach

Purpose: The Pandemic Covid 19 has changed the learning and teaching activities where effective teaching styles are being explored to enhance the transfer of learning. The aim of this case study is to focus on students' understanding level to capture and store contents learned in their memory specifically on the university’s code subject, the Fundamentals of Entrepreneurship. Design/Methodology/Approach: The application of the Mnemonic Keyword Method in the teaching and learning activities. It encodes the information so that it can be more easily retrieved and the sentences are transformed into catchy and simple formulas. The students were asked to participate in the online brain testing platform to determine their brain dominance and applied the relevant pedagogical strategy, the Mnemonic KeyWord Approach in the teaching settings. The https://braintest.sommer-sommer.com/en/ testing tool or platform is applied to differentiate between the left and right brains amongst the selected students. Findings: Results revealed a strong effect of knowledge retention and the teaching tools based on the SUFO (Students Online Feedback) evaluation. Results indicate that the method is effective because it provides a meaningful formula in terms of coding, and abbreviation formulas generation. Implications/Originality/Value: It is concluded that the keyword approach is an effective strategy for memorising and retaining knowledge. It encodes the information so that it can be more easily retrieved and the sentences are transformed into catchy and simple formulas.

Description of survival data extended to the case of competing risks: a teaching approach based on frequency tables

Survival analysis is a powerful statistical tool to study failure-time data. In introductory courses students learn how to describe right-censored survival time data using the product-limit estimator of the survival function on a given end-point relying on a product of conditional survival probabilities. In the case of a composite end-point, the next step is to account for the presence of competing risks. The complement to one of the survival function is decomposed into the sum of cause-specific incidences, which are obtained as sum of unconditional probabilities due to the single competing risk. However, this algebraic decomposition is not straightforward, given the difference between the structure of the involved estimators. In addition, one is tempted to use the Kaplan-Meier estimator, leading to an erroneous decomposition of the overall incidence. Here we discuss a simple reinterpretation of the Kaplan-Meier formula in terms of sum of non-conditional probabilities of developing the end-point in time, adjusted for the presence of censoring. This approach could be used for describing survival data through simple frequency tables which are directly generalized to the case of competing risks. In addition, it makes clear how the estimation of the single cause-specific incidence through the Kaplan-Meier estimator, simply considering the occurrence of competing events as censored data, leads to an overestimation of the cause-specific incidence. Two examples are provided to support the explanation: the first one, could help to clarify the procedure described by the formulas; the second one, simulates real data in order to present graphically the results.

Effectiveness, Sensitivity, and Specificity of Intraocular Lens Power Calculation Formulas for Short Eyes.

Objectives:To compare intraocular lens (IOL) power calculation formulas in terms of absolute error (AE) and receiver operating characteristic curves in eyes with axial length (AL) shorter than 22.0 mm.Materials and Methods:The data of hyperopic patients who underwent uneventful phacoemulsification with IOL implantation in MW-med Eye Centre, Cracow, Poland between October 2015 and June 2019 were retrospectively reviewed. IOL power was calculated using Holladay1, SRK/T, Hoffer Q, Holladay2, Haigis, and Barrett Universal II formulas. The power of the implanted lens was based on Hoffer Q. Three months after phacoemulsification, refraction was measured and AE was calculated. The percentage of patients with full visual acuity without any correction and the percentage of hyperopic patients was determined for each formula. Receiver operating characteristic curves with cut-off points for AL were drawn for each formula and the area under the curve was evaluated.Results:Fifty-six patients (62 eyes) whose ocular AL ranged between 20.58 mm and 21.97 mm were included in the study. Hoffer Q formula yielded the lowest mean AE (0.09±0.08 D), the highest percentage of patients with full visual acuity without correction (75.8%), and the lowest rate of postoperative hyperopia (8.1%). However, the SRK/T formula had the largest area under the curve (0.667).Conclusion:The Hoffer Q formula gave the lowest level of AE in the study and seems to be recommendable for IOL power calculation for hyperopic eyes. Further studies are needed on the use of receiver operating characteristic curves in assessing the effectiveness of IOL power calculation formulas.

Curve counting via stable objects in derived categories of Calabi-Yau 4-folds

In our previous paper with Maulik, we proposed a conjectural Gopakumar-Vafa (GV) type formula for the generating series of stable pair invariants on Calabi-Yau (CY) 4-folds. The purpose of this paper is to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category. We introduce invariants counting LePotier's stable pairs on CY 4-folds, and show that they count certain stable objects in D0-D2-D8 bound states in the derived category. We propose a conjectural wall-crossing formula for the generating series of our invariants, which recovers the conjectural GV type formula. Examples are computed for both compact and toric cases to support our conjecture.

On the non-critical exceptional zeros of Katz p-adic L-functions for CM fields

The primary goal of this article is to study p-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of vanishing of Katz p-adic L-functions associated to CM fields, by means of a leading term formula we prove in terms of Rubin–Stark elements. In the particular case when the CM field in question is imaginary quadratic, our leading term formula and its consequences we record herein are unconditional.

Unique temperature distribution and explicit efficiency formula for one-dimensional thermoelectric generators under constant Seebeck coefficients

A thermoelectric generator converts a temperature difference into electrical energy. Its energy conversion efficiency is determined by the steady-state temperature distribution inside the generator. By assuming the thermoelectric material in the generator has a temperature-independent Seebeck coefficient and the generator is one-dimensional, we show that the second-order integro-differential equation describing the inside temperature distribution has a unique solution for any given ratio of external load resistance to the internal resistance. Hence the efficiency is well defined. Furthermore, we show the efficiency has an explicit formula in terms of the temperature-dependent thermal conductivity and electrical resistivity of the thermoelectric material. On the contrary, the integro-differential equation may have multiple solutions if an external load resistance value is given instead of the external-load-to-internal resistance ratio.

Effect of combination of some Polymers with Carbopol 940 on Pregabalin Release Rate from Emulgels

Topical Emulgels of Pregabalin were formulated using Carbopol 940 in combination with (PVP k90, PEG 200, PEG 1000, HPMC K15M) to obtain the optimal formula in terms of drug release rate. Physical appearance, pH, viscosity, stability, drug content, In-vitro drug release, Differential scanning calorimetry and Fourier transform infrared spectroscopy, were tested in all formulas and evaluated to determine the best one. All formulations had good physical properties and stability, F2 which contains Carbopol 940 0.4% and HPMC K15M 0.4% gave the best drug release rate: (30% in 20 min and 93% in 360 min). But F3 which contains Carbopol 940 0.4% and PEG 200 10% showed the less release rate at 360 min: (64%). F2 showed Korsmeyer–Peppas model Kinetic release. So F2 was the best formula because it released drug since 10 min and still to 360 min (6 hours), thus F2 can be used to obtain rapid analgesic effect and avoid CNS-mediated side effects of Pregabalin.

Performance of the CUSUM Control Chart Using Approximation to ARL for Long- Memory Fractionally Integrated Autoregressive Process with Exogenous Variable

The cumulative sum (CUSUM) control chart is a well-known process monitoring tool that is sensitive to small-to-moderate changes in process parameters. In this paper, we propose an approximated average run length (ARL) method based on the numerical integral equation (NIE) method for monitoring the mean of a long-memory autoregressive fractionally integrated process with an exogenous variable (ARFIX) running on a CUSUM control chart. The approximated ARL based on the NIE method is realized by solving a system of linear equations and integration based on the partitioning and summation of the area under the curve of a function derived by using the Gauss-Legendre quadrature. In a comparative study with the ARL based on analytical formulas, the proposed approximated ARL method could detect shifts of various sizes in the process mean of an ARFIX process running on a CUSUM control chart. In addition, the proposed method was compared with their analytical formulas in terms of the relative percentage change (r%) to verify the accuracy of the ARL results. The results revealed that the ARL results obtained from the NIE method are an approach to analytical formulas with r% of less than 0.25. Hence, the NIE method is very accurate and in excellent agreement with the analytical formulas approach. Apparently, the NIE method is an alternative as efficiently as the analytical method for this situation. It also performed well in comparison with the approximated ARL for the same process running on an exponentially weighted moving average control chart. In addition, real datasets are also used to demonstrate the efficacy of the proposed method.

Casting light on shadow Somos sequences

AbstractRecently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the $\mathbb{C}$ -algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for $f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when $\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as an application, we construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

An experimental measurement research on impact load of high speed water jet acting on a rigid target by a Hopkinson pressure bar

A high speed water jet generation device based on explosion in a cylindrical cavity is introduced, which can eject a well-characterized water jet with a velocity around 100 m/s. The evolution characteristics and impact phenomenon of water jet act on a rigid target are observed by a high speed camera and the impact load is measured by a Hopkinson pressure bar. The top shape of water jet develops from a convex spherical surface to a flat plane as the stand-off increases under the influence of air resistance. When the top of water jet contacts the target plate, a transient high-pressure zone appears at the liquid-solid interface. A horizontal annular splashing jet generates and the velocity of splashing jet is much higher than that of incident jet as the result of high-pressure zone. The impact load is measured by the Hopkinson pressure bar, including transient “water hummer” pressure and subsequent hydrodynamic load. The “water hummer” pressure of water jet with a flat top is twice that with a convex spherical top, while the subsequent hydrodynamic pressure is equivalent at a comparable transient impact velocity. The measurement results by a Hopkinson pressure bar are in good agreement with the theory formulas in terms of both “water hummer” pressure and hydrodynamic pressure. It is certified that the Hopkinson pressure bar has a credible measuring capacity for both transient and steady-state pressure. By controlling cavity depth, charge amount and stand-off distance, the different top shape and velocity of water jet can be obtained.

Topology optimization for a water-cooled heat sink in micro-electronics based on Pareto frontier

• An efficient Pareto frontier based design strategy for heat sinks is proposed. • A fitting formula of the Pareto frontiers is introduced to accelerate the design. • The 2D topology design results are validated by 3D CFD simulations. • Effects of temperature dependent properties on the results are significant. Topology optimization has been increasingly adopted as a non-intuitive methodology for heat sink design in the thermal management of microelectronics. In this work, a bi-objective topology optimization method for heat sink design in micro-electronics is suggested by considering the trade-off between two contradicting objectives, i.e., the heat transfer and hydraulic performances. The method in this work focused on the laminar water flow with Reynolds number less than 100 based on the channel width for a wide temperature range that covers different types of micro-electric devices. A series of Pareto frontiers were acquired for heat sink designs with different topologies, which can be used to decide the optimum design in respect of heat transfer and pressure drop. Particularly, an analytical formula in terms of inlet Reynolds numbers was obtained for the fast generation of the Pareto frontiers under different working conditions which help to determine the weight parameters of two contradicting objectives readily. The new strategy for weight parameter determination significantly reduces the workload of trial-and-error during the weight parameter determination in the traditional bi-objective topology optimization and thus accelerates the design process. Besides, the temperature-dependent physical properties (viscosity and thermal conductivity) in the design model were considered and were found to affect the design results significantly. Finally, the 3D heat sink model was reconstructed from the 2D topology optimization, and the thermal-fluid analyses were conducted. The results indicate that the 3D reconstructed model can well satisfy the design objective specified in the topology optimization, thereby confirming the validity and robustness of our design method.

Hecke action on the principal block

In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical$p$-Kazhdan–Lusztig polynomials.

Profiling Growth Patterns of Human Gut Microbes in Response to Preterm Human Milk and Formula

BackgroundThe infant gut microbiota undergoes significant fluxes in microbial composition during the first few years of life. The intestine is colonized shortly after birth with facultative anaerobes, like as Enterococcus, Enterobacter, Streptococcus, and Staphylococcus, which reduce the intestinal oxygen content and create niches for strict anaerobes, like Bifidobacteria, Bacteroides, and Clostridia. The dominant factors which influence the infant gut composition are gestational age and diet. The gut microbiota of preterm infants is characterized by limited microbial diversity and increased levels of facultative anaerobes such as Enterococci, Staphylococci, and Enterbacteriacea (Enterobacter, Escherichia, and Klebsiellaspp.). The gut microbiota is also highly influenced by diet, specifically formula and human milk use. The intake of human milk is associated with increased Bifidobacteria, while the intake of formula correlates with increased Enterobacteriaceae. This study aimed to analyze next‐generation products including preterm formula, human milk‐oligosaccharide term formula, and preterm breastmilk using a culture‐based model.Methods &amp; ResultsWe cultured 8 different Bifidobacteria, 8 Lactobacilli, 8 Streptococcus, 3 Enterococci and 4 Klebsiella strains in a fully defined bacteria media and monitored growth and pH after 20 hrs of anaerobic incubation. Significant differences were identified in strain growth and culture pH between human milk and formula. Human milk significantly elevated the growth of all Bifidobacteria and Lactobacilli. In contrast, human milk supported the growth of three Streptococci and only nominally supported pathobiont Enterococci and Klebsiella strains. Preterm formula elevated the majority of Bifidobacteria and Lactobacilli, as well as several Streptococci. Interestingly, preterm and full‐term formula significantly enhanced the growth of pathobiont strains. Full‐term formula only supported the growth certain commensal microbes, which was in marked contrast to human milk and preterm formula. Using computational modeling, a correlation was observed with human milk oligosaccharide (HMO) degrading glycosyl hydrolase profiles in the genomes of commensal microbes with human milk, but not with formula; confirming the HMO preference of our representative infant commensal microbes.ConclusionsThese findings indicate a unique profile of growth in response to human milk and formula. Since alterations in the infant gut microbiota have linked to childhood health and development, understanding how bacteria differentially utilize whole milk or formula may provide insights into which microbes drive diet‐related microbiota shifts.