Practical Formula Research Articles

Remarks on the Coefficients of Inverse Cyclotomic Polynomials

Cyclotomic polynomials play an imporant role in discrete mathematics. Recently, inverse cyclotomic polynomials have been defined and investigated. In this paper, we present some recent advances related to the coefficients of inverse cyclotomic polynomials, including a practical recursive formula for their calculation and numerical simulations.

Exploring the Feasibility of Calculating Expected pCO2 From Venous Blood Gas Samples Alone in Intensive Care Patients.

Introduction This study highlights the significance of assessing acid-base balance and gas exchange in intensive care patients. The research investigates the applicability of using the "expected (pCO2 = HCO3 + 15)" formula, derived from venous blood gas samples, as an alternative to Winter's formula and practical formula. The study emphasizes the importance of identifying the primary acid-base abnormality accurately and efficiently for appropriate clinical intervention in critically ill patients. Methods This study included 400 adult patients admitted to the Anesthesia Clinic in the Third Stage Anesthesia and Reanimation Intensive Care Unit at Hitit University Erol Olçok Training and Research Hospital between April 2020 and July 2023. Blood gas samples were collected simultaneously from both arterial lines and venous catheters. Patients under 18 years, pregnant women, hemodialysis patients, and those with missing data were excluded. The study aimed to calculate the expected partial pressure of carbon dioxide (pCO2) values using Winter's formula and simple formulafor both arterial and venous blood gas samples and assess potential correlations between them. Results The results showed a narrow range for arterial pH values (7.12-7.72), a wider distribution for pCO2 values (17.90-81.30 mmHg), and a moderate dispersion for HCO3values (12.80-44.33 mmol/L). Both Winter's and simple formulas were applied to estimate the expected pCO2 values, showing strong positive correlations between arterial and venous pH, pCO2, and HCO3 values. The scatterplot illustrated a very high level of association (Pearson's correlation coefficient, r = 1) between the expected pCO2 values derived from both formulas using arterial and venous blood gas samples. Conclusion The clinical study demonstrates that estimating expected pCO2 values in mixed acid-base disorders can be achieved using a simple and convenient formulation, eliminating the need for arterial blood gas sampling and its associated complications.

Simplified Formula for Estimating Nasal Dimensions for 3-Dimensional Facial Reconstruction among Japanese Adults

The eyelids, external nose, and lips play an important role in individual identification and facial recognition; however, they are excluded from tissue marker points, and are reconstructed based on generic methods for 3D facial reconstruction or facial approximation. Therefore, this study focused on nasal dimensions and evaluated whether Krogman’s widely used formula for estimating the dimensions of an external nose, regardless of sex, race, and body physique, can be applied to Japanese adults. A total of 146 postmortem CT images of Japanese adult cadavers (64 males and 82 females, aged 58–105 years old) were retrospectively analyzed. The total nasal projection (TNP) among Japanese adults was estimated using the formula, TNP = 1.9 × the anterior nasal spine projection (ANSP) + the mid-philtrum depth (MPD), which differed significantly from the coefficient (3.0) in the conventional formula, regardless of sex, race, and body physique, and therefore needed modification for Asians. Although there was no positive relationship between the total nasal width (TNW) and the maximum width of the anterior nasal aperture (ANAW), the TNW could be estimated by adding soft tissue that varies by sex and body physique to both sides of the nearly constant ANAW. Therefore, we determined a simple and practical formula to estimate nasal dimensions among Japanese adults for conventional 3D facial reconstruction and manual 3D facial sculpture.

The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a basis‐free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalized to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems ofanalysis of evolution of the entangled quantum states.

Prediction of flexural resistance of reinforced concrete beam cross-section by practical formula

This paper presents the process of developing a practical formula for predicting the ultimate bending moment of rectangular reinforced concrete (RC) beams through regression analysis. The data used for regression analysis was generated by using the fiber method to analyze a non-linear batch of commonly encountered RC beam cross-sections. The practical formula was obtained by fitting a linear regression model to the training set and then making predictions on the test set. The coefficient of determination, R2, between the bending moment values calculated from the formula and the results of the non-linear analysis was 0.9948, indicating a good predictive capability of the formula.

More precise measurement of irregular splenic volume in cirrhotic patients with portal hypertension

Splenomegaly often occurs in cirrhotic patients with portal hypertension (PHT), and therefore, the efficacy and accuracy of conventional methods for measuring splenic volume are a matter of question in these patients. Here, we developed a novel approach to assess true splenic volume more precisely. High-quality thin-slice computed tomography data of 112 cirrhotic patients with PHT were obtained and reviewed. Both the conventional measurement and a novel formula obtained from 3-dimensional reconstruction software were used to estimate splenic volume, and the accuracy was compared and verified. In PHT patients, the splenic volume calculated using the conventional method was significantly less than that calculated using the 3-dimensional software. We found that the splenic volume was significantly positively correlated with splenic indices of length (L), thickness (T), and width (W) and also the diameter of the splenic vein. Using these indices, we propose 2 novel formulas using the software to estimate the splenic volume more accurately: SV = 69.686 × L + 53.077 × W + 103.525 × T + 314.510 × diameter of splenic vein −2266.209 (p < 0.01, R2 = 0.805). And a more practical simplified formula: SV' = 0.504 × L × W × Τ + 319.762 × diameter of splenic vein −81.66 (p < 0.01, R2 = 0.784). Although the conventional formula has been widely used for years, it is not suitable for an enlarged spleen. We developed 2 novel formulas for estimating splenic volume from clinical data that were more appropriate for cirrhotic patients with PHT.

Gold nanoparticles and breast cancer: A bibliometric analysis of the current state of research and future directions

Gold nanoparticles (GNPs) are widely searched for their usage in breast cancer research because of their unique features. For example, these particles can deliver drugs to specific sites, making imaging and photothermal therapy possible, thus rendering them suitable particles for theranostic purposes. Bibliometric research is a statistical analytical technique useful for the data systematically found in the literature. In this bibliometric study, the global research output regarding GNPs in breast cancer (BC) research was analyzed, mapped, and evaluated using bibliometric indicators.All documents related to the application of GNPs applications in BC research that were published in English-language, peer-reviewed journals over the past 20 years were retrieved from the Scopus database. Bibliometric indicators were extracted using VOSviewer and Biblioshiny. Thematic maps, conceptual maps, and visualization graphs were created.A total of 2035 published documents were retrieved. Chinese authors were the most active, having published approximately 27.5% of the total documents, while researchers from the United States (USA) had the most significant scientific impact. The study revealed weak international collaboration in this research area. Keywords mapping identified the main research themes and hotspots in this research field. Multimodal approaches in cancer treatment and diagnostics, targeted and effective cancer treatment, aptamers and biosensors, and green synthesis of GNPs were the four clusters retrieved from theme and hotspot analyses. This study analyzed and mapped the expanding field of GNPs in breast cancer research (GNP-BCR) and identified various applications of GNPs in this field. GNPs were used as drug delivery systems to target specific cancer cells and improve anticancer drug bioavailability. Different treatment and diagnostic modalities were revealed, such as photodynamic therapy (PDT) and photothermal therapy (PTT). Moreover, the development of aptamer-based biosensors using gold nanoparticles was identified as a niche theme in this research, while the green synthesis of GNPs emerged as a new and promising theme. However, clinical research is still warranted to translate this fundamental knowledge into practical and human-useable formulas.

Buckling capacity of steel circular hollow section considering the restraint of welded hollow spherical joints

An integral finite element model of steel circular hollow section and welded hollow spherical joints is established to consider the restraint of welded hollow spherical joints on steel circular hollow section in this paper. The buckling modes are used as the initial geometrical imperfection forms of the steel circular hollow section. Based on the statistics method, a method for calculating the buckling capacity design value of the steel circular hollow section with uniform reliability considering the restraint of welded hollow spherical joints is established, and the formula for calculating error of this value at certain confidence is deduced. The relationships between the buckling capacity design value of steel circular hollow section and the structural parameters of the member and joint are determined quantitatively by linear regression method. A practical formula for the buckling capacity design value of steel circular hollow section considering the restraint of welded hollow spherical joints is derived. Compared to the results calculated by the design code indicates that if the difference of restraint effect which is caused by different welded hollow spherical joint is ignored, the buckling capacity value of steel circular hollow section calculated based on the unified effective length coefficient may be bigger or smaller than that of the actual case. Therefore, refined consideration should be placed on the different restraint of welded hollow spherical joints of different sizes on steel circular hollow section, and the restraint of the end joints should be accurately considered when determining the buckling capacity of steel circular hollow sections.

Assessment on effective stiffness of RC hollow columns based upon semi-static experiment

The effective stiffness is a key parameter to assess the seismic performance and supply rational design for a column of the regular concrete structures. Due to the detail difference between the reinforced concrete (RC) hollow and solid columns, the applicability of the existing models, almost obtained from the test results of solid columns, to the hollow members should be studied deeply. The rationality of only a few available formulas from hollow specimens should be validated through experimental investigation. Therefore, the test data of 28 rectangular hollow columns in literature were selected to evaluate the existing equations for effective stiffness. To explore the nonlinear characteristics of RC hollow columns and supply more practical and rational formulas of stiffness, 7 rectangular and 5 round-end hollow specimens were tested by pseudo-static loading to clarify the critical factors of effective stiffness. The analysis on test data shows that the effective stiffness increases almost linearly with increased axial compression and shear span ratio. The quantity of rebar, strain penetration, strength of concrete and steel have little influence on effective stiffness compared with the axial compression and aspect ratio. A novel simplified formula was proposed based on the foregoing 35 rectangular hollow specimens considering the effect of axial pressure ratio, shear span ratio. The rationality and generality of the suggested formulas were verified according to the test results of five round-ended hollow specimens. The comparison analysis shows that the proposed model is more applicable than the existing formulas to assess the effective stiffness of different types of hollow columns with more precise.

A Formula of Solutions for Non-Autonomous Linear Difference Equations With a Fractional Forward Operator

Abstract In this article, we define a fractional forward discrete operator. Then, for a family of linear nonautonomous fractional difference equations constructed by using this fractional discrete operator, we provide a practical formula of solutions. This family of problems covers several linear fractional difference equations that appear in the literature. Numerical examples are given to justify our theory.

Numerical Simulation of the Dynamic Responses and Impact-Bearing Capacity of CFDST Columns under Lateral Impact

The concrete-filled double steel tube (CFDST) has been more and more widely used in infrastructure due to its greater moment of inertia compared to the ordinary concrete-filled steel tube (CFST). With the increase in collision accidents occurring in recent years, the research on CFDST under impact needs more attention. To study the dynamic responses of CFDST columns under lateral impact, the finite element package ABAQUS was used to simulate the impact force and mid-span deflection of the column under the coupling action of axial force and lateral impact based on the comparison of experimental data in the literature. The influence of hollow ratio, axial compression ratio, slenderness ratio, impact energy, and material strength on the dynamic responses of the CFDST column are studied. The results show that with the increase in the hollow ratio from 0.3 to 0.8, the residual mid-span deflection of the CFDST column decreases by 77.9%. The mid-span deflection shows an increasing trend following a decreasing one with the axial compression ratio increasing. With the increase in the slenderness ratio from 30 to 60, the plateau impact force decreases by 62.5%, and the residual mid-span deflection increases by 240.5%. The practical formulas for the impact-bearing capacity and residual mid-span deflection of the CFDST column are suggested with high accuracy.

A THEORETICAL APPROXIMATION FOR LAMINAR FLOW BETWEEN ECCENTRIC CYLINDERS

Taylor-Couette flow between two concentric cylinders has received much attention due to its use in various applications, including biomedical devices, micro electro-mechanical systems, polymer pumping and electric motor cooling. Due to the complex interaction of the viscosity and the involved geometry within the confined space, different flow regimes are dominant under different conditions, affecting the fluid dynamics and heat transfer. In analyzing the mentioned flow, besides the experimental and computational studies, analytical models have been developed with varying levels of complication. In the present study, using the homotopy of both the flow and the domain geometry between the concentric and eccentric cylinders, a practical formula for flow between eccentric cylinders is developed. In doing so, an appropriate transformation function for the geometry is developed and embedded into the velocity equation for the concentric cylinders. The resultant equation is tested against flow simulation results. A validity margin analysis is performed based on the variation of the mass flow rate between the cylinders. It is seen that the proposed model for eccentric cylinders is applicable for all gap distances, unlike the previous models that are restricted to narrow gaps. Finally, a separate formula to quantify the error in the estimates of the present method is also derived, which involves the ratio of the cylinders and the eccentricity.

Experimental Investigation on the Performance Levels and Drift Capacity of SRC Columns

To promote the performance-based seismic design and assessment of steel-reinforced concrete (SRC) structures, 27 SRC columns, varying axial load ratio n, steel ratio ρss, and shear span ratio λ, were tested under quasi-static cyclic loadings. The failure modes, hysteretic response, performance levels (PLs), damage states and drift limits of the SRC columns were reported and discussed. The results showed that the failure mode shifted gradually from flexural to shear as λ decreased, but the deformability was not weakened obviously. Six PLs were defined based on the backbone curve of the SRC columns. Each PL was correlated to a damage state that represents a specific macroscopic damage extent. The relative error of drift limits between performance levels and damage states was less than 9% on average. Parametric analysis showed that n and ρss were the most influential factors for drift limits, whereas λ had an ignorable effect. Then, practical regression formulas were established in terms of n and ρss for drift limits of three key PLs (PL1, PL5, and PL6) of the SRC columns, and they were in good agreement with the experimental results. Finally, the average damage indices of PL1 to PL6, calculated using the model proposed in this paper, were 0.26, 0.45, 0.65, 0.83, 0.95, and 0.99, respectively.

PRACTICAL EVALUATION FORMULA OF ELASTIC BUCKLING CAPACITY FOR CEILING BRACE ELASTICALLY SUPPORTED ON MIDPOINT

Brace is usually installed in aseismic ceiling to keep the aseismic capacity. In the case of high depth ceiling, brace crosses hanging bolt. Experience of several tests show qualitatively that buckling capacity become larger than that of Euler buckling by combining the brace to hanging bolt. Evaluating formula for buckling capacity of brace supported by spring at midpoint is derived and its practical formula is proposed, in which supporting spring ratio ξ and capacity amplifying ratio η are introduced. Buckling capacity of 3 case of lateral loading test results show good agreement with evaluated values by proposed formula.

Compressive performance of welded hollow spherical joints with external triangular ribs

This study addresses the axial compressive performance of welded hollow spherical (WHS) joints with external triangular ribs. External triangular ribs are able to postpone the local buckling of the joint during axial loading, and thus strengthen the compressive capacity of it. A simplified analytical model is proposed to elucidate the stress distribution on the hollow sphere of axial-loaded WHS joints with external triangular ribs. Then a simplified solution for the yield strength of axial-loaded WHS joints is deduced based on the elastic thin shell theory. The simplified solution indicates that the yield strength of WHS joints with external triangular ribs can be expressed by a coefficient related to the specifications of the ribs and the joints multiply by that of WHS joints without external ribs. Compression tests on 6 full-scale specimens of WHS joints with varied configurations of triangular ribs are conducted. It shows that the failure mode and the strength enhancement of the specimens vary as the specification of the triangular ribs change, and the triangular ribs enhance the strength of the joints up to 77 %. A refined FE model is established and calibrated against the findings from the experiments. Parametric studies on the design parameters which may affect the compressive response of the WHS joints with triangular ribs are fully investigated. It is shown that the compressive yield strength of the joint rises with the increase of the number, height and thickness of the triangular ribs, and decreases with the increase of the angle-of-elevation of the triangular ribs. It is also shown that the effect of the ratio of the diameter of the sphere to that of the steel tube on the compressive yield strength of the joint is much lighter than the others, and deemed as negligible. Regression analysis on the experimental and numerical data is carried out to obtain a practical formula for the compressive capacity of WHS joints with external triangular ribs. This formula is incorporated with the previously developed formula for the compressive capacity of WHS joints with external trapezoidal ribs, to form a general formula applicable for WHS joints with either external triangular ribs or external trapezoidal ribs.

A practical empirical formula for the calculation of ship added friction-resistance due to (bio)fouling

An accurate prediction of the ship added friction-resistance due to (bio)fouling is important in the maritime sector in view of environmental sustainability. For industrial needs, the prediction method must not only be accurate but also be practical. An empirical formula is proposed in this study for the calculation of ship added friction-resistance due to (bio)fouling. An optimization method was applied, in which the objective function is defined as the least mean square error between the predicted results and data derived from Demirel et al.'s diagrams. The mathematical model to be fitted with the data involves power functions whose form resembles the functional relationship between the added friction-resistance coefficient and Reynolds number, and between the added friction-resistance coefficient and roughness ratio. The coefficients, powers and constant in the mathematical model were determined by using a nonlinear generalized reduced-gradient method. Comparisons of the formula predictions with CFD calculation results for a flat plate and the Kriso Container Ship show a small percentage difference of less than 5%. Further, the formula predictions differ less than 12% from the results obtained from methods utilizing topographical roughness parameters available in the literature.

QTc interval measurement in patients with right bundle branch block: A practical method.

Prolonging the QT interval in the right bundle branch block (RBBB) can create challenges for electrophysiologists in estimating repolarization time and eliminating the effect of depolarization changes on QT interval. In this study, we aimed to develop a practice formula to eliminate the effect of depolarization changes on QT interval in patients with RBBB. This prospective study evaluated accidentally induced RBBB in patients undergoing electrophysiological study. Two expert electrophysiologists recorded the ECG parameters, including QRS duration, QT interval, and cycle length, in the patients. The formula was developed based on QT interval differences (with and without RBBB) and its proportion to QRS. Additionally, the Bazzet, Rautaharju, and Hodge formulas were used to evaluate QTc. We evaluated 96 patients in this study. The mean QT interval without RBBB was 369.39 ± 37.38, reaching 404.22 ± 39.23 after inducing RBBB. ΔQT was calculated as 34.83 ± 17.61, and the ratio of ΔQT/QRS with RBBB was almost 23%. Our formula is: (QTwith RBBB - 23% × QRS). Subtraction of 25% instead of 23% seems more straightforward and practical. Our formula could also predict the QTc interval in RBBB based on the Bazzet, Rautaharju, and Hodge formulas. Previous formulas for QT correction were hard to apply in the clinical setting or were not specified for RBBB. Our new formula allows a rapid and practical method for QT correction in RBBB in clinical practice.

Compressive performance of an innovative tall pier with composite box section

To improve the seismic performance of tall-pier bridges, an innovative tall pier with a composite box section, composed of four-limb concrete-filled steel tube (CFST) columns and low-yield-strength (LYS) steel plates, was proposed based on the principle of capacity design and the replaceable fuse concept. Both axial and eccentric compression tests were conducted on a series of specimens to study the compressive performance of the composite tall pier. Parametric analysis was performed to investigate the effects of load eccentricity, slenderness ratio, thickness, and yield strength of the LYS steel plates on the compressive capacity using validated finite element (FE) models. The main results indicate that the composite tall pier fails with global bending deformation and elastoplastic instability under axial or eccentric compression. The compressive capacity and initial stiffness of the composite tall pier decrease significantly with an increase in eccentricity or slenderness ratio, and the influences of eccentricity and slenderness ratio are essentially independent. Additionally, the increase in the thickness of the steel plates enhanced the flexural stiffness and compressive capacity, and reduced the uneven loading distribution for the pier. Under axial compression, the pier can be simplified as a CFST lattice column connected by steel beams and LYS steel plates. Finally, practical formulas for the compressive capacity are proposed considering the connection and restraint effects of the LYS steel plates on the CFST columns, which can be used to predict the compressive capacity of the novel composite tall pier.

Renoprotective Effect of Isoorientin in Diabetic Nephropathy via Activating Autophagy and Inhibiting the PI3K-AKT-TSC2-mTOR Pathway.

Diabetic nephropathy (DN) is one of the most serious complications of diabetes and the most common cause of death. The autophagy of podocytes plays an important role in the pathogenesis of DN. Here, through screening the constituent compounds of practical and useful Chinese herbal formulas, we identified that isoorientin (ISO) strongly promoted the autophagy of podocytes and could effectively protect podocytes from high glucose (HG)-induced injury. ISO significantly improved autophagic clearance of damaged mitochondria under HG conditions. Through a proteomics-based approach, we identified that ISO could reverse the excessive phosphorylation of TSC2 S939 under HG conditions and stimulate autophagy through inhibition of the PI3K-AKT-TSC2-mTOR pathway. Furthermore, ISO was predicted to bind to the SH2 domain of PI3Kp85[Formula: see text], which is crucial for the recruitment and activation of PI3K. The protective effect of ISO and its effects on autophagy and particularly on mitophagy were further proved using a DN mice model. To summarize, our study identified the protective effects of ISO against DN and demonstrated that ISO was a strong activator of autophagy, which could provide a basis for drug development.

An abstract Lagrangian framework for computing shape derivatives

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.