Univariate Equation Research Articles

Novel Weakness Multivariate Quadratic Structures Detected within Macaulay Matrix

The security of a Multivariate Public-Key Cryptosystem (MPKC) is based on the hard mathematical problem of solving Multivariate Quadratic (MQ) equations over finite fields, also known as the MQ problem. An MPKC has the potential to be a post-quantum cryptosystem. In this paper, we identify new weaknesses in the Macaulay matrix identified via Wang's technique, which was initially designed for solving multivariate quadratic equation systems. This new weakness occurs in the case of random coefficients in any column vector for different variables of monomials and random coefficients are assigned to other monomials. The weakness is exposed through the use of Gaussian elimination to obtain a univariate equation. We illustrate our findings using a random example.

Low-Complexity Beamforming Designs and Channel Estimation for Passive-Intelligent-Surface-Assisted MISO Energy Transfer

The usage of passive intelligent surface (PIS) is emerging as a low-cost green alternative to massive antenna systems for realizing high energy beamforming (EB) gains. Considering the limited computational capability and constant-envelope precoding for PIS, we propose three novel low-complexity passive EB designs for optimizing the efficacy of PIS-assisted energy transfer (PET) from a multiantenna power beacon (PB) to a single-antenna energy harvesting (EH) user. The first EB design involves solving a univariate equation, and closed forms expressed are presented for the other two. Further, to maximize the practical utility of PET, we introduce a novel channel estimation (CE) protocol for obtaining least-squares estimators for the channels as required for EB designing. Using them, we also derive closed-form expressions for optimal PIS location and optimal time allocation between CE and PET within each coherence block to maximize the user’s net harvested energy. Numerical results verify the CE analysis and validate the novel analytical bound derived for received power during PET and proposed PIS designs’ quality against existing benchmarks. We show that the proposed jointly optimal design for PET can yield a significant improvement of about 15dB, and a reduced active array size at PB can achieve the desired EB gain with sufficient passive elements at PIS. Lastly, we also briefly discuss how the proposed CE and EB designs can be extended to the multiuser settings.

Transient Hydromagnetic Stagnation-Point Flow across a Vertical Surface with Partial Slip in a Chemically Reactive Medium

A numerical study of partial slip boundary condition is investigated. The stagnation-point flow problem involving some physio-chemical parameters has been elucidated. The process involves developing a multivariate mathematical model for the flow and transforming it into a coupled univariate equation. Key parameters of interest in the study are the buoyancy force, the surface stretching, the unsteadiness, the radiation, the dissipation effects, the slip effects, the species reaction and the magnetic field parameters. It is concluded that the impact of physio-chemical factors significantly alters the kinematics of the flow in order to optimally achieve desired product characteristics.

How Does the Gradient Measure of the Lung SBRT Treatment Plan Depend on the Tumor Volume and Shape?

PurposeGradient measure (GM) is a critical index related to normal tissue sparing in radiosurgery. This study aims to describe the dependence of GM on target volume and target shape for lung stereotactic body radiation therapy (SBRT) treatment plans.MethodsA total of 307 peripheral and 119 central lung SBRT treatment plans were enrolled for this study. A least-squares regression was used for data analysis. First, the equations with different functional forms were established to determine the dependence of GM on a univariaty (VP or Sp) and bivariaty (VP and Sp), respectively. Then, the correlation coefficients and p-values of variables for all equations were compared and analyzed to determine the dependence of GM on PTV volume (VP) and sphericity (Sp).ResultsThe power equations had the highest coefficient of determination (R2) in the dependence results of GM on univariate VP. The equations were and for peripheral and central lesions, respectively. On the other hand, the R2 of all functional forms were less than 0.25 when the relationship of GM versus univariate Sp was analyzed. Similarly, the power equation also obtained the highest R2 in bivariaty VP and Sp analysis, whether for central or peripheral. However, the R2 of the bivariate equations were not improved compared with those of univariate equations. Moreover, the p-values of the variable Sp were greater than 0.05.ConclusionsThe GM of the lung SBRT plan is shape-independent and volume-dependent. The dependence of GM on PTV volume for peripheral and central lung cancer can be described by two different power equations. The results of this study can be used as a potential tool to assist dosimetric quality control during the radiosurgery process.

Estimation of large dimensional time varying VARs using copulas

This paper provides a simple, yet reliable, alternative to the (Bayesian) estimation of large multivariate VARs with time variation in the conditional mean equations and/or in the covariance structure. The original multivariate, n-dimensional model is treated as a set of n univariate estimation problems, and cross-dependence is handled through the use of a copula. This makes it possible to run the estimation of each univariate equation in parallel. Thus, only univariate distribution functions are needed when estimating the individual equations, which are often available in closed form, and easy to handle with MCMC (or other techniques). Thereafter, the individual posteriors are combined with the copula, so obtaining a joint posterior which can be easily resampled. We illustrate our approach using various examples of large time-varying parameter VARs with 129 and even 215 macroeconomic variables.

Efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian

This paper focuses on efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian. Based on the Lagrangian duality theory, we deal with the projection problem via a semismooth Newton algorithm with line search safeguard, which admits global and locally quadratic convergence, to solve a univariate semismooth equation. Numerical experiments show that our proposed algorithm outperforms favourably the existing state-of-the-art standard solvers and is able to reliably solve very large-scale projection problems. Besides, we derive an explicit expression of a generalized Jacobian of the studied projection, which is an essential component of second-order nonsmooth methods.

In Brief

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

A Multicenter Study to Evaluate Harmonization of Assays for C-Terminal Telopeptides of Type I Collagen (\xdf-CTX): A Report from the IFCC-IOF Committee for Bone Metabolism (C-BM)

BackgroundBiochemical bone turnover markers are useful tools to assess bone remodeling. C-terminal telopeptide of type I collagen (ß-CTX) has been recommended as a reference marker for bone resorption in research studies.MethodsWe describe the results of a multicenter study for routine clinical laboratory assays for ß-CTX in serum and plasma. Four centers (Athens GR, Copenhagen DK, Liege BE and Sheffield UK) collected serum and plasma (EDTA) samples from 796 patients presenting to osteoporosis clinics. Specimens were analyzed in duplicate with each of the available routine clinical laboratory methods according to the manufacturers’ instructions. Passing-Bablok regressions, Bland–Altman plots, V-shape evaluation method, and Concordance correlation coefficient for ß-CTX values between serum and plasma specimens and between methods were used to determine the agreement between results. A generalized linear model was employed to identify possible variables that affected the relationship between the methods. Two pools of serum were finally prepared and sent to the four centers to be measured in 5-plicates on 5 consecutive days with the different methods.ResultsWe identified significant variations between methods and between centers although comparison results were generally more consistent in plasma compared to serum. We developed univariate linear regression equations to predict Roche Elecsys®, IDS-iSYS, or IDS ELISA ß-CTX results from any other assay and a multivariable model including the site of analysis, the age, and weight of the patient. The coefficients of determination (R2) increased from approximately 0.80 in the univariate model to approximately 0.90 in the multivariable one, with the site of analysis being the major contributing factor. Results observed on the pools also suggest that long-term storage could explain the difference observed with the different methods on serum.ConclusionOur results show large within- and between-assay variation for ß-CTX measurement, particularly in serum. Stability of the analyte could be one of the explanations. More studies should be undertaken to overcome this problem. Until harmonization is achieved, we recommend measuring ß-CTX by the same assay on EDTA plasma, especially for research purposes in large pharmacological trials where samples can be stored for long periods before they are assayed.

Tools for analyzing the intersection curve between two quadrics through projection and lifting

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

Correlation between chemical composition, EHGE and TME of corn for ducks

Correlations between chemical composition, enzymatic hydrolysate gross energy (EHGE), and true metabolizable energy (TME) of corn for ducks were investigated. Twenty-two corn samples were collected from various regions in 11 provinces of China. The crude protein (CP), ether extract (EE), neutral detergent fibre (NDF), Ash, gross energy (GE), dry matter (DM), amylopectin (AP), amylose (AM), total starch (TS), and AP/AM were determined for each sample. Five of the samples of corn were chosen at the mean, mean ± 1 standard deviation (SD), and mean ± 2 SD based on AP/AM. The EHGE of these samples was analysed using the pepsin-artificial small intestinal fluid enzymatic method. These five samples were also force-fed to male Cherry Valley ducks to assay their TME. Finally, correlation analyses were performed, and regression equations were established. Ash content, GE, and TS were highly related to EHGE. Univariate prediction equations were EHGE = 11.8566Ash-0.0421 (P <0.05), EHGE = 0.1535GE1.5642 (P <0.05), and EHGE = 0.1020TS1.1561 (P <0.05). The total starch, AP/AM, and ash of the chemical compositions were highly related to TME. The corresponding univariate regression equations were TME = 21.9355TS-0.0910 (P <0.05), TME = 15.6590AP/AM-0.0559 (P <0.05), and TME = 15.0778Ash0.0442 (P <0.05). The mean EHGE was equivalent to 78.5% of TME, but their correlation coefficient was low. In conclusion, chemical composition was predictive of EHGE and TME of corn samples for ducks, but the correlation of EHGE and TME was low
 Keywords: Cherry Valley duck, amylopectin, amylose, true metabolizable energy

Symbolic Proof of Bistability in Reaction Networks

Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to determine, exclusively via symbolic computations, the stability of the steady states for unspecified parameter values. In particular, our approach fully determines the stability type of all steady states of a broad class of networks. To this end, we combine the Hurwitz criterion, reduction of the steady state equations to one univariate equation, and structural reductions of the reaction network. Using our method, we prove that bistability occurs in open regions in parameter space for many relevant motifs in cell signaling.

Analysis of Cavitation Processes in Xylem

Cavitation in plants is caused by development of air bubbles, which is related to their equilibrium and development. There is a univariate cubic equation for bubble balance. New root formula of this kind of equation was proposed by Shenjin Fan, which is simpler than the Caldan’s. Using Shenjin formulas and taking water pressure P1 as an independent variable, this paper gives the exact solution of the equation under certain conditions. The stability of the equilibrium of an air bubble in its different radius ranges is obtained by the way different from the previous. This kind of cavitation includes two types: First type may be caused by the growth of pre-existent air bubbles; Second type is air seeding, here defined as the sucking of air bubbles from already gas-filled conduits. For air seeding three ways of cavitation have been proposed. For the first type this paper puts forward that two ways of cavitation can occur, which are the same with the first two ways of air seeding except of air reservoirs. Moreover, for the first way of the first type, the range of water pressures is the same with that of the first way of air seeding. For the second way of the first type the range of water pressures is much wider, or the pressure range equals the pressure sum of the second and third ways of air seeding. Through the specific data the relationship between the two types is given.

A self-tuning algorithm to approximate roots of systems of nonlinear equations based on the firefly algorithm

The most acquainted methods to find root approximations of nonlinear equations and systems; numerical methods possess disadvantages such as necessity of acceptable initial guesses and the differentiability of the functions. Even having such qualities, for some univariate nonlinear equations and systems, approximations of roots is not possible with numerical methods. Research are geared towards finding alternate approaches, which are successful where numerical methods fail. One of the most disadvantageous properties in such approaches is inability of finding more than one approximation at a time. On the other hand these methods are incorporated with algorithm specific parameters which should be set properly in order to achieve good results. We present a modified firefly algorithm handling the problem as an optimisation problem, which is capable of giving multiple root approximations simultaneously within a reasonable state space while tuning the parameters of the proposed algorithm by itself, using a self-tuning framework. Differentiability and the continuity of the functions and the close initial guesses are needless to solve nonlinear systems using the proposed approach. Benchmark systems found in the literature were used to test the new algorithm. The root approximations and the tuned parameters obtained along with the statistical analysis illustrate the viability of the method.

A Fast Inverse Scattering Imaging Method by Applying Virtual Experiments to T-Matrix Scheme

As a non-iterative linear inverse scattering imaging method, the linear sampling method (LSM) has the merits of easy implementation and high computational efficiency. In this article, based on the LSM and T-matrix method, we propose a fast method to reconstruct the target medium parameters. First, we use the LSM qualitative imaging method to estimate the contour of the target. Then, the preprocessing information of the LSM is applied to the virtual experiment circles in the framework of T-matrix method. In this way, we avoid the inversion of the matrix and only need to solve a series of univariate equations. Numerical simulation results show that the reconstruction accuracy of the proposed method is similar to that of LSM quantitative imaging method, but it has higher calculation accuracy and wider application range. Although the calculation accuracy is not as good as the contrast source inversion (CSI) method, the proposed method has huge advantages in terms of calculation efficiency.

New Selective Harmonic Elimination-Pulse-Width Modulation for Cascaded H-bridge Multilevel Inverters

AbstractThis paper presents a univariate equation-based selective harmonic elimination-pulse-width modulation (USHE-PWM) method for cascaded H-bridge (CHB) multilevel inverters (MLIs). Using equal-area criteria, the selection of voltage rating of dc-links have been done. The output voltage is controlled by inserting switching notch at the centre of each level of output voltage. Instead of conventional transcendental SHE-PWM equations, all these calculations are accomplished using univariate equations. This simplifies the real-time implementation of the proposed method greatly, even for the MLI with a large number of levels. The USHE-PWM method can eliminate more harmonics compared with the conventional SHE-PWM methods. While controlling the output voltage, the harmonic pattern remains unaffected. The validity of the proposed method is confirmed by the simulation and experimental studies.

Nondimensionalized Univariate Equation Characterizing Optimal State Feedback Control for Collision Avoidance

In this paper, integration of steering and braking are studied for collision avoidance from the standpoint of optimal control theory. First, the collision avoidance by braking, steering, and steering with braking maneuvers are theoretically investigated. By some assumptions, such as, the point mass vehicle model and constant resultant vehicle force in all directions, the selection of the best maneuver for collision avoidance is represented by the regions on a 2-D diagram. The nondimensionalized axes of this diagram stand for all required vehicle state and geometrical conditions, more specifically, the remaining distance, longitudinal vehicle velocity, lane width, and friction coefficient. Second, a two-point boundary value problem derived for the integrated control is reduced to a single equation with an unknown via algebraic simplifications, and the bisection method turns out to be the most appropriate among the algorithms that have been tested. It turned out that the efficient and precise solution of the optimal control problem is guaranteed, including the cases in which the intervention has been initiated. Consequently, this paper realizes the efficient identification of the theoretically reliable reference maneuver of single lane change with the minimum collision avoidance distance in the framework of feedback control.

Prediction of the length of the femur from its fragments in a Sri Lankan population

ABSTRACTIn many forensic circumstances, the femur is fragmented due to decomposition and mutilation, thus necessitating development and application of regression equations to estimate the length of the femur from its fragments. In this study, regression equations were derived for the prediction of the femoral length, utilizing 11 standard femoral measurements (maximum length of femur, epicondylar breadth, vertical head diameter, transverse head diameter, vertical neck diameter, transverse neck diameter, neck length, subtrochanteric anterior–posterior diameter, subtrochanteric transverse diameter, mid-shaft anterior–posterior diameter and mid-shaft transverse diameter) obtained from 86 femora belonging to a contemporary Sri Lankan population. Univariate regression equations with moderate to high correlations (0.226 to 0.714) were formulated for the estimation of the maximum femoral length from the measurements of fragments of the femur. Of the single variables, the transverse neck diameter emerged as the best predictor (r = 0.714, SEE = 18.6596) of the maximum length of the femur. The multiple regression equations using a combination of variables yielded higher correlations (0.774–0.857) with lower error estimates (14.18–16.96). The regression equations presented in this study can be used to estimate the length of the femur of unknown, mutilated and dismembered human skeletal remains with considerable accuracy.

The salary of physicians in Chinese public tertiary hospitals: a national cross-sectional and follow-up study

BackgroundSalary is perceived as a key factor affecting job satisfaction, employment, and the retention or migration of physicians within and across countries. This study aimed to describe physicians’ salary and workload and to examine the factors determining the salary of physicians in China.MethodsWe conducted a self-administered, smartphone-based national survey in 136 tertiary hospitals across 31 provinces. A total of 17,615 physicians self-reported their salaries after tax and their characteristics, practice settings, and work efforts in 2015. Then, 2498 of the physicians were followed up for the second-round survey in 2016. Univariate analysis and general estimate equations were applied to evaluate the factors associated with salary.ResultIn 2015, the average annual salary of the physicians was US$13,764. Physicians in eastern China earned more than those in central (p < 0.001) and western China (p = 0.002) after adjustment for locality expenditure per capita. The salary for men in 2015 was US$14,832, which was more than that for women (US$12,912; p < 0.001). Of the respondents, 76.3% worked more than 40 h per week. The physicians dealt with 40 patients per day on average. Consequently, 67.2% physicians spent no more than 10 min with each outpatient. After adjustments for age and management position, salary was associated with years in practice, education background, and specialty, but not with weekly work hours and gender.ConclusionsThe physicians’ salaries were relatively low, and the majority of the respondents worked more than 40 h per week. Years in practice, education background, specialty, and region were associated with salary, while weekly work hours and gender were not. To better remunerate Chinese physicians, more resources are demanded, and a workload-based salary scheme should be adopted.

New Approach in Assessing Core Muscle Endurance Using Ratings of Perceived Exertion

George, JD, Tolley, JR, Vehrs, PR, Reece, JD, Akay, MF, and Cambridge, EDJ. New approach in assessing core muscle endurance using ratings of perceived exertion. J Strength Cond Res 32(4): 1081-1088, 2018-This study sought to develop regression models to estimate maximal endurance time using data from 4 core muscle endurance tests. Eighty healthy university students (age: 22.7 ± 1.9 years) performed the plank, right side-bridge, left side-bridge, and back extension tests in a random order. Participants were instructed to hold each static position for a maximal endurance time, while maintaining proper form, and then rest for 5 minutes between tests. A test administrator recorded participants' ratings of perceived exertion (RPE; a modified 10-point scale) every 5 seconds. Based on regression analysis, the elapsed time to reach an RPE of 8 (RPE8) exhibited statistical significance (p < 0.0001) and the highest accuracy as compared with lower RPE values. The following univariate regression models were generated to estimate maximal endurance time across the 4 tests: plank (r = 0.94; standard error of estimate [SEE] = 17.6 seconds; n = 77) = 23.9 + (1.110 × RPE8); right side-bridge (r = 0.92; SEE = 11.4 seconds; n = 80) = 18.5 + (1.022 × RPE8); left side-bridge (r = 0.93; SEE = 10.8 seconds; n = 80) = 16.8 + (1.062 × RPE8); and back extension (r = 0.93; SEE = 14.2 seconds; n = 79) = 21.5 + (1.027 × RPE8). These results suggest that submaximal protocols based on elapsed time to reach RPE8 provide strength and conditioning professionals relatively accurate univariate regression equation estimates of maximal core muscle endurance time and offer a viable submaximal alternative to maximal capacity testing when time efficiency, participant safety, or certain educational objectives may be a priority.

Regional division and reduction algorithm for minimizing the sum of linear fractional functions

This paper presents a practicable regional division and cut algorithm for minimizing the sum of linear fractional functions over a polyhedron. In the algorithm, by using an equivalent problem (P) of the original problem, the proposed division operation generalizes the usual standard bisection, and the deleting and reduction operations can cut away a large part of the current investigated region in which the global optimal solution of (P) does not exist. The main computation involves solving a sequence of univariate equations with strict monotonicity. The proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of univariate equations. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.