Order Polynomial Research Articles

Hot Deformation Behavior of Inconel 718 Alloy Using Regression-based Dynamic Material Model and Applications in Ring Rolling Manufacturing

Inconel 718 nickel-based alloy is extensively used in the aerospace industry (e.g., gas turbine engine components) because of its excellent corrosion resistance and high mechanical properties at elevated temperatures. However, there is a certain limit to manufacturing the alloy through plastic deformation due to its high deformation resistance and complicated deformation behaviors. In this study, the hot deformation behavior of Inconel 718 alloy was investigated to establish how processing conditions of flow stress-strain, at strain rates from 0.001 to 10 s<sup>-1</sup>, and temperatures from 850 to 1200<sup>o</sup>C, affected dynamic recrystallization. The regression-based material model was utilized to calculate the strain-rate sensitivity, and subsequently depict the efficiency of the power dissipation and instability criterion of hot deformation. The processing map and instability criterion predicted by the developed 3<sup>rd</sup>-order polynomial regression model corresponded with the experimental results and in particular, showed a better prediction for instability regime compared to the existing discrete derivative approach. Predicting the strain-rate sensitivity values on a continuous scale with regression analysis covered the additional instability region of the high strain rate near 10 s<sup>-1</sup>. The dynamic recrystallization deformation was also characterized by microstructural analysis along with the processing map. Consequently, ring-rolled aviation parts were manufactured with the optimum processing parameters, which conform to the AMS 5663 standard (Aerospace material specifications for Inconel 718).

A new interpolation equation in the ITS-90 subrange from the triple point of water to the freezing point of indium

For all the subranges above 0.01 °C, other than the interpolation between the triple point of water (TPW) to the melting point of gallium, temperatures on the international temperature scale of 1990 (ITS-90) are specified without the melting point of gallium. This study suggests a new interpolation equation in the subrange from the TPW to the freezing point of indium which is a polynomial of fractional third order with two coefficients to be determined by the measuring values of standard platinum resistance thermometers (SPRTs)’ resistance ratios at the melting point of gallium (W Ga) and at the freezing point of indium (W In). A sample set of 30 SPRTs from various countries and laboratories is used to quantify the comparisons between this new interpolation and the ITS-90. Analyses show that the reproducibility is significantly improved compared with the ITS-90 in the same subrange. The peak value of overlapping subrange inconsistencies (SRI) from TPW to the melting point of gallium is reduced from 0.91 mK to 0.21 mK. Both the mean and standard deviation of the SRI peak values for the new equation decrease by approximately a factor of four compared to those of the ITS-90 (mean decreasing form 0.20 mK to 0.05 mK and standard deviation decreasing form 0.32 mK to 0.07 mK). Type 3 non-uniqueness (NU3) determinations of the same subrange have also been looked into with a data set of four SPRTs compared in thermostatic baths using a copper block and the results show that NU3 scales down from a range −0.15 mK–0.25 mK of the ITS-90 to −0.10 mK–0.15 mK of the new interpolation equation. Propagation of Uncertainties investigation shows that the new equation inflates merely 7% the measurement uncertainties of fixed points in a relatively short interval from 30 °C to 60 °C.

Workload Comparison of Contemporary Interval Throwing Programs and a Novel Optimized Program for Baseball Pitchers.

In the rehabilitation of injured baseball pitchers, there is lack of consensus on how to guide a player back to pitching. It is unknown how different contemporary interval throwing programs (ITPs) progress in the amount of throwing workload. To 1) evaluate three prominent ITPs commonly employed in baseball pitcher rehabilitation and assess whether these ITPs produce training loads that increase in a controlled, graduated manner and 2) devise an ITP that produced training loads which increased steadily over time. Cross-sectional study. Three publicly available ITPs from prominent sports medicine institutions were analyzed. Elbow varus torque per throw was calculated from a 2nd order polynomial regression based upon a relationship between recorded torque measurements and throwing distance measured from a database of 111,196 throws. The relative rate of workload increase was measured as an acute:chronic workload ratio (ACWR). For each ITP, throw counts, daily/acute/chronic workloads, and ACWR were calculated and plotted over time. Finally, an original ITP was devised based upon a computational model that gradually increases ACWR over time and finished with an optimal chronic workload. Each ITP exhibited a unique progression of throwing distances, quantities, and days to create different workload profiles. The three ITPs had throwing schedules ranging from 136 days to 187 days, ACWR spiked above or fell below a literature-defined "safe" range (i.e.0.7 - 1.3) 19, 21, and 23 times. A novel ITP, predicated on a 146-day schedule and with a final chronic workload of 14.2, was designed to have no spikes outside of the safe range. Existing ITPs widely utilized for rehabilitation of baseball pitchers exhibit significantly inconsistent variation in the rate of throwing load progression. Computational modeling may facilitate more incremental workload progression in ITPs, thereby reducing injury during rehabilitation and more efficiently condition a pitcher for return to competition. 3b.

A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.

New bright and dark stochastic optical solitons related to an eighth-order NLSE in the presence of higher order polynomial nonlinearity

New bright and dark stochastic optical solitons related to an eighth-order NLSE in the presence of higher order polynomial nonlinearity

Genetic parameters for novel mastitis traits defined by combining test-day somatic cell score and differential somatic cell count in the first lactation of Japanese Holsteins

In this study, we aimed to improve current udder health genetic evaluations by addressing the limitations of monthly sampled somatic cell score (SCS) for distinguishing cows with robust innate immunity from those susceptible to chronic infections. The objectives were to (1) establish novel somatic cell traits by integrating SCS and the differential somatic cell count (DSCC), which represents the combined proportion of polymorphonuclear leukocytes and lymphocytes in somatic cells and (2) estimate genetic parameters for the new traits, including their daily heritability and genetic correlations with milk production traits and SCS, using a random regression test-day model (RRTDM). We derived 3 traits, namely ML_SCS_DSCC, SCS_4_DSCC_65_binary, and ML_SCS_DSCC_binary, by using milk loss estimates at corresponding SCS and DSCC levels, thresholds established in previous studies, and a threshold established from milk loss estimates, respectively. Data consisted of test-day records collected during January 2021 through March 2022 from 265 herds in Hokkaido, Japan. From these records, we extracted records between 7 to 305 d in milk (DIM) in the first lactation to fit the RRTDM. The model included the random effect of herd-test-day, the fixed effect of year-month, fixed lactation curves nested with calving age groups, and random regressions with Legendre polynomials of order 3 for additive genetic and permanent environmental effects. The analysis was performed using Gibbs sampling with Gibbsf90+ software. The averages (ranges) of daily heritability estimates over lactation were 0.086 (0.075 to 0.095) for SCS, 0.104 (0.073 to 0.127) for ML_SCS_DSCC, 0.137 (0.014 to 0.297) for SCS_4_DSCC_65_binary, and 0.138 (0.115 to 0.185) for ML_SCS_DSCC_binary; the heritability curve for SCS_4_DSCC_65_binary was erratic. Genetic correlations within the trait decreased as the DIM interval widened, especially for those integrating DSCC, indicating that these traits should be analyzed using RRTDM rather than repeatability models. The averages (ranges) of genetic correlations with milk yield over lactation were 0.01 (-0.22 to 0.28) for SCS, -0.05 (-0.40 to 0.13) for ML_SCS_DSCC, -0.08 (-0.17 to 0.09) for SCS_4_DSCC_65_binary, and -0.08 (-0.22 to 0.27) for ML_SCS_DSCC_binary. Compared with SCS, the newly defined traits exhibited slightly stronger negative genetic correlations with milk yield. Especially in late lactation stages, the genetic correlation between ML_SCS_DSCC and milk yield was significantly below zero, with a posterior median of -0.40. Furthermore, the new traits showed positive correlations with SCS, having estimates varying from 0.68 to 0.85 for ML_SCS_DSCC, 0.14 to 0.47 for SCS_4_DSCC_65_binary, and 0.61 to 0.66 for ML_SCS_DSCC_binary, depending on DIM. Considering that ML_SCS_DSCC and ML_SCS_DSCC_binary have relatively high heritability (compared with SCS) and favorable genetic correlations with milk production traits and SCS, their incorporation into breeding programs appears promising. Nevertheless, their genetic relationships with (sub)clinical mastitis require further investigation.

Platelet Modeling in DHF Patients Using Local Polynomial Semiparametric Regression on Longitudinal Data

Regression analysis is one of the statistical methods used to model the relationship between response variables and predictor variables. Semiparametric regression is a combination of parametric and nonparametric regression. The estimator used in estimating the semiparametric regression model in this research is the Local Polynomial. Longitudinal data can be found in the health sector, including dengue hemorrhagic fever (DHF) data. The laboratory criteria for indication of DHF is thrombocytopenia. This research aims to obtain platelets model for DHF patients that can be used for forecasting so that it is hoped that it can provide information to the medical team in treating DHF patients. The estimated model used is Local Polynomial semiparametric regression on longitudinal data. The response variables in this research were platelets of DHF patients, which were influenced by hemoglobin as parametric predictor variable and examination time while hospitalized as nonparametric predictor variable. In the local polynomial regression model, it is necessary to select the optimal bandwidth and polynomial order method, GCV. The optimum bandwidth selection based on the GCV method obtained is 1.5 and polynomial order of 2, then applied to DHF patient platelet data, producing an estimated local polynomial semiparametric regression model that follows the actual data pattern. Modeling the platelets of DHF patients obtained using a local polynomial estimator resulted in an R2 value of 84.25% and MAPE of 4.5%, indicating highly accurate forecasting, so it can be concluded that the resulting model is better at predicting.

Evaluating dose coverage and conformity in stereotactic ablative body radiotherapy (SABR) plans

PurposeAccepted conformity metrics in stereotactic ablative body radiotherapy (SABR) have significant limitations. This work aimed to develop a spatial assessment methodology that improves and automates checks of dose prescription and dose gradient from planning target volume (PTV) edge. MethodsA Python-based script was developed to determine linear distances from the PTV edge to specified isodose, every 15 degrees on all axial slices and along the central axis in the coronal plane. A new “Internal PTV contour” distance metric is introduced as a size and shape indicator. 134 previously treated SABR patients stratified by anatomical site and PTV volume were analysed to establish baselines and tolerances for automation acceptability. ResultsIn the axial plane, median distance (MD) from PTV edge to the 100 % isodose was 0.13 mm (range: −0.67 to 0.53 mm), and for the 90 % isodose was 2.37 mm (1.36 to 3.40 mm). Lung and non-Lung dose gradient criteria was established by fitting a second order polynomial to the MD as a function of “Internal PTV contour”. This resulted in acceptability criteria of MD + 1 mm for 80 % isodose and MD + 2 mm for the 50 % isodose. For the coronal plane, MD to the 100 % isodose was 0.49 mm (−1.24 to 2.14 mm) and for the 90 % was 1.73 mm (−0.49 to 4.13 mm). ConclusionsOur in-house script enables a high-quality spatial assessment of PTV dose coverage and gradient, with the new ‘Internal PTV contour’ distance metric correlating well with dose gradient.

Choosing the Best Functions for the Rayleigh-Ritz Vibration Analysis of Beams

The Rayleigh-Ritz Method is used extensively for the vibration analysis of structures. The accuracy depends on the assumed functions. In this work several different groups of functions are examined and compared for the accuracy of the resulting natural frequencies, and for the overall mode shape error norms calculated with respect to the known exact solutions. It is concluded that a set that combines low order polynomials, odd cosine and odd sine functions, or, even cosine and even sine functions, is more likely to yield the best accuracy and convergence of both frequency and mode shapes for a general beam structure.

Analytical method for stable background reduction for Raman spectra of carbon‐containing meteorite and terrestrial samples suffering from intense fluorescence

AbstractChemical states of carbon in terrestrial (meta) sediments and carbonaceous chondrites gather attention as a geothermometer. As a nondestructive analytical method, Raman spectroscopy has been widely used to study their electronic properties, crystallinity, and structural defects through so‐called D and G bands. For the analysis of Raman spectra, a common problem is coexistence of a fluorescence background, which should be subtracted prior to the peak‐fitting analysis. However, we recently faced a problem that the band shape noticeably changed depending on the background function assumed although the background seemed to be well subtracted at a first glance regardless of the choice of the background function. For the application of the Raman spectroscopy as a geothermometer, a standard background subtraction method must be established to suppress the arbitrariness. In the present study, Raman spectra of seven carbon‐containing natural samples, whose background intensities were significantly different, were measured, and their background shape was evaluated by first‐, second‐, and third‐order polynomials. The results indicated that the third‐order polynomial was necessary and sufficient as a standard background function. Importantly, although lower order polynomials seem to successfully fit the background at a first glance, they falsely caused dispersion of the shoulder band shape.

Ultrasound-assisted extraction of anthocyanin from black rice bran and its encapsulation by complex coacervation

The present work was planned to optimize extraction process of phenolics, flavonoids, monomeric anthocyanin and antioxidants from black rice bran using Ultrasound-assisted extraction and finally utilize it for encapsulation using double emulsion coacervation process. Response surface methodology (RSM) was employed for optimization of extraction process. Acidified ethanol (1 M HCl) was used as extraction solvent and ultrasound power (W) and extraction time (min) were used as independent variables. Face centered composite design (FCCD) data was successfully opted and developed a 2nd order polynomial equation with R2 value of 0.94 for total phenolic content (TPC), 0.95 for total flavonoid content (TFC), 0.87 for 1,1-diphenyl-2-picrylhydrazyl (DPPH) and 0.98 for total anthocyanin content (ANCs) respectively. Kinetic model study for extraction of anthocyanin was compared to conventional extraction process. Characterization of the extracts was performed using high pressure liquid chromatography analysis (HPLC). Cyanidin-3-glucoside is the predominated anthocyanin found as compared to peonidin-3-D-glucoside. Finally, coacervated microcapsules were developed using anthocyanin extract as core solution and gelatin and acacia gum as wall materials. Characterization of the microcapsules in terms of moisture content, hygroscopicity, solubility, encapsulation efficiency and surface morphology were evaluated. The result thus obtained shows that potential anthocyanin content from black rice bran can be utilized as food ingredients for various food application and formulation of functional food.

Calculation of the detection limits with the least-squares method in gamma-ray spectrometry: A simple procedure

A metrologically consistent procedure for assessing the detection limits of activity measurements for gamma-ray emitters with high-resolution spectrometers using the LSQ method is described and tested. As the input to the assessment, besides the measured contents of the spectral channels, the results of the peak analysis, i.e., the indication and its uncertainty, are used. The unfolding of the spectral region of interest into its components corresponding to the peak representing the indication and its background allows us to take into account the uncertainty budget, describing the uncertainty of the indication and the shape of the corresponding peak, making possible to include these sources of uncertainty in the calculation of the decision threshold. To assess the detection limit, the variance of the indication is calculated as a function of the indication itself, while considering the relative uncertainty of the conversion factor. The variance of the indication observed is approximated by a polynomial of the second order of the indication, thus making it possible to calculate the detection limit analytically. The method was tested on measured spectra using the empirically determined spectral shape of the peak representing the indication. It was shown how the empirically determined shape of an isolated and expressive peak close to the peak representing the indication can be used in the calculation of the decision threshold and how the presence of a peak overlapping with the peak representing the indication affects the decision threshold and the detection limit. It is explained that besides the counting statistics, the sources of uncertainty due to the shape of the peak representing indication also contribute to the decision threshold. However, to the increase of the detection limit over the decision threshold, besides the counting statistic, only the uncertainty of the conversion factor contributes. It is shown that in the presence of the indication, the decision threshold and the detection limit can be used to quantify the comparison between the observed value and the true value of the measurand with a predetermined quantity value in terms of the probabilities of making errors of the first and second kind. The application of the decision thresholds and detection limits to a conformity assessment is proposed.

Prediction of photovoltaic waste generation in Canada using regression-based model.

The global surge in photovoltaic (PV) installations and the resulting increase in PV waste are a growing concern. The aims of this study include predicting the volume of photovoltaic waste in Canada. The forecasting of solar waste volume employed linear regression, 2nd order polynomial regression, and power regression models. The study's results indicate that Canada is on the verge of facing challenges related to the end-of-life treatment of photovoltaic modules in the coming years due to the significant growth in PV capacity over recent decades. According to the analysis, for early loss, the PV waste volume in 2045 could range from 180,000 MT to 270,000 MT, and for regular loss, it could range from 160,000 MT to 180,000 MT. This research is anticipated to assist relevant government agencies in assessing the prospective volume of PV waste to establish a sustainable and resilient PV waste management plan for Canada. These findings may shed light on the feasibility of a circular economy and advocate for the involvement of all stakeholders in a carefully coordinated strategy to mitigate potential environmental impacts and optimize resource utilization efficiency.

Optical, electrical, and thermal performance enhancement for a concentrating photovoltaic/thermal system using optimized polynomial compound parabolic concentrators

Performance of a concentrating photovoltaic/thermal system directly depends on the solar irradiation gathered by concentrators. Few studies tackled the systemic and general optimizations of optical efficiency and uniformity of energy distribution for two-dimensional compound parabolic concentrators (CPCs). In this paper, a polynomial CPC is proposed. To evaluate the optical efficiency and uniformity of the concentrated irradiation, an interpolation method is also proposed. Using genetic algorithm, the optimized polynomial CPCs with different polynomial orders are determined. With fourth order, the concentrating efficiency reaches the theoretic limit, and the concentrated irradiation is relatively uniform. Accordingly, the optimized polynomial CPCs for concentration ratios not larger than 10 are determined. Using a double-pass spectral beam splitting photovoltaic/thermal system model, the electrical and thermal performance using the determined optimized polynomial CPCs is calculated. For concentration ratios not larger than 10, the optimized polynomial CPC is able to provide high concentrating efficiencies with good uniformity. Moreover, the maximum temperature of the photovoltaic cell is smaller than 325 K. And then, the optimized polynomial CPCs for multiple incident angles are presented, along with the system performance. For multiple incident angles, the optimized polynomial CPC also shows good performance, but the size increases.

Succinylated starch emulsified Eugenol and Carvacrol nanoemulsions improved digestive stability, bio-accessibility and Salmonella typhimurium inhibition

The ultrasonically processed Eugenol (EU) and Carvacrol (CAR) nanoemulsions (NE) were successfully optimized via response surface methodology (RSM) to achieve broad spectrum antimicrobial efficacy. These NE were prepared using 2 % (w/w) purity gum ultra (i.e., succinylated starch), 10 % (v/v) oil phase, 80 % (800 W) sonication power, and 10 min of processing time as determined via RSM. The second order Polynomial method was suitable to RSM with a co-efficient of determination >0.90 and a narrow polydispersity index (PDI) ranging 0.12–0.19. NE had small droplet sizes (135.5–160 nm) and low volatility at high temperatures. The EU & CAR entrapment and heat stability (300 °C) confirmed by Fourier transform infrared spectroscopy (FTIR) and thermogravimetric analysis (TGA). Further, the volatility of EU & CAR NE was 18.18 ± 0.13 % and 12.29 ± 0.11 % respectively, being lower than that of bulk/unencapsulated EU & CAR (i.e., 23.48 ± 0.38 % and 19.11 ± 0.08 %) after 2 h at 90 °C. Interestingly, both EU & CAR NE showed sustained release behaviour till 48 h. Their digest could inhibit Salmonella typhimurium (S. typhimurium) via membrane disruption and access to cellular machinery as evident from SEM images. Furthermore, in-vivo bio-accessibility of EU & CAR in mice serum was up to 80 %. These cost-effective and short-processed EU/CAR NE have the potential as green preservatives for food industry.

Contactless in-vivo temperature sensing in spinal cord based on magnetically bistable microwire

We have employed a magnetically bistable glass coated microwire as a passive contactless sensing element for in-vivo temperature monitoring in the rat’s spinal cord. The single piece (1 cm long) of Fe76Cr2Mo6B15Cu1 microwire was epidurally implanted into the adult female Wistar rats. The magnetization switching was measured in-vivo within the temperature range of 34.5–41.0 °C. Results show a monotonous temperature decrease of the switching time whose fitting to a second order polynomial dependence allows one to conclude a sensitivity down to 0.01ºC within the measured interval. Biocompatibility study shows that cells have a high affinity to the glass-coated microwire, and their viability is not influenced by the experimental condition under which the microwire’s response is measured. Finally, a histology study reveals that no histopathological changes or tissue reactions were observed even after 9 months of implantation.

Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions

Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order <i>α</i> in terms of hyperbolic functions

On Stirling and bell numbers of order 1/2

The Stirling numbers of order 1/2 (of the second kind) introduced by Katugampola are discussed and it is shown that they are given by a scaled subfamily of the generalized Stirling numbers introduced by Hsu and Shiue. This allows to deduce in a straightforward fashion many properties of the Stirling and Bell numbers of order 1/2, for example, recurrence relations, generating functions, Dobi?ski formula, and Spivey formula. The even Bell polynomials of order 1/2 are shown to be closely related to generalized Laguerre polynomials of order ?1/2. Generalized Stirling numbers of order 1/2 of the first kind are defined and studied. An analog of the Weyl algebra is introduced and proposed as a natural algebraic setting where the Stirling numbers of order 1/2 of both kinds appear as ordering coefficients. This algebra contains the Weyl algebra as a subalgebra.

Toroidal seawater conductivity sensors and instrumentation for anti-submarine warfare

This paper describes the development of toroid-based seawater conductivity sensors and instrumentation for 0 to 60 mS/cm conductivity range and 50 bar hydrostatic pressure operational capability which is equivalent to 500 metres of ocean depth. Double core toroidal transformer concept was used and investigated for Mn-Zn and Fe based nanocrystalline cores, turns ratio and temperature dependence. Different excitation and sense coil turn numbers were fabricated by keeping the turn&#x2019;s ratio 1:2, 1:4, 1:6 and 1:8. Optimum frequency of Mn-Zn based conductivity sensors with lower and higher turn numbers were observed for turn ratio 1:8 and 1:4 respectively at 120 kHz and obtained voltage responses of 952 and 1567 mV. Fe based Nano crystalline core with turn ratio 1:8 shows an optimum frequency of 70 kHz and giving a voltage difference of 830 mV. Two dimensional polynomials of order 2 were presented for temperature dependence of conductivity for Mn-Zn core based sensors. Sensor responses of Fe based nanocrystalline core sensors in &#x2018;lower turn number&#x2019; configuration with same turn ratio is presented and observed linear temperature dependence compared to Mn-Zn based sensors. The mechanical design and frequency of operation of developed conductivity sensors gives inherent immunity to ASW band of frequency interference. The efficacy of conductivity sensor with measuring electronics and commercial standard sensors profiling was proved experimentally compared up to 500 metres in sea profiling and obtained a root mean square error of 0.42 for profile data.

Random walks on a line and algebraic curves

This work is devoted to the studying the generating function of the first hitting time of the positive semi-axis under the homogeneous discrete integer random walk on a line. In the first part of the work the increments are supposed to be independent. Recurrent relations for the probabilities allow us to write the system of equations for the sought generating function. Applying the resultants technique, we succeed to reduce this system to a single equation. Then we can study it by calculating the genus of the correposponding plane algrebraic curve via analyzing its singularities. In the work we write the sought equations for some random walks and we show that if the increments take all integer values from $-2$ to $2$, or from $-1$ to $3$ with equal probabilities or they take equally probable values $-1$ and $4$, then the curve is rational, while this is not true in the first case. In the second part of the work we consider a symmetric process, the increments take the values $-1$, $0$, $1$, but then we suppose a non-zero correlation of each next increment with the previous one. For such process the equation for the generating function defines an elliptic curve depending on the square of the correlation coefficient for neighbouring increments if all increments are non-zero and it defines a hyperelliptic curve of genus $2$. The degeneration criterion of the latter is the presence of multiple roots of a sixth order polynomial under general symmetrically distributed conditional probabilities.