Second-degree Polynomial Equation Research Articles

On the relationship between human all-cause mortality and age.

Many equations can be used to study the relationship between mortality rates and age: Gompertz, Weibull, logistic, polynomial and age-period-cohort equations a.o. All these equations result in highly significant correlations between ln mortality rates and age in the age range 35-84 years. This applies to all developed countries and is independent of the differences in causes of death between populations. The best fit is obtained by a second-degree polynomial equation (R2 > 0.99), closely followed by the Gompertz equation. This equation is preferred in view of its extreme simplicity. A highly significant correlation exists between the intercept and the slope of the Gompertz equations, pointing to a crossing-over age. Beyond that age, around 85 years, populations with high mortality rates have a lower mortality, due to selective survival of the strongest individuals. The polynomial age 2 term may be positive or negative, an expression of the acceleration or de-acceleration of mortality at higher ages and is significantly more often positive in women. The equations used are very useful for the study of the aging process and for examining the relationship between possible causal factors and mortality rates in populations.

Prediction of the Hiestand Bonding Indices of Binary Powder Mixtures from Single-Component Bonding Indices

ABSTRACTA priori predictions of the bonding indices of binary powder mixtures from the single-component indices were attempted. The binary mixtures were classified according to the mechanism of deformation of the single-components as plastic–plastic, brittle–brittle, and plastic–brittle. When the components of a binary mixture consolidated by the same mechanism (plastic–plastic and brittle–brittle mixtures), a straight line could be fit to a plot of bonding index versus composition. This linearity indicates that the bonding index can be reliably estimated by interpolation between the two single-component bonding indices. When the mixtures were such that one component was brittle while the other was plastic, a linear function did not fit the data. For these cases, a second-degree polynomial equation could befit to a plot of bonding index versus composition. A combination of multiple linear regression and trial and error was used to generate a single generalized equation. For pairs of compounds wherein each compound has a different compaction mechanism, this new equation appears to allow satisfactory predictions of the bonding indices of mixtures with varying compositions using only the single-component bonding indices.

An empirical method for estimating travel times for wet volcanic mass flows

Travel times for wet volcanic mass flows (debris avalanches and lahars) can be forecast as a function of distance from source when the approximate flow rate (peak discharge near the source) can be estimated beforehand. The near-source flow rate is primarily a function of initial flow volume, which should be possible to estimate to an order of magnitude on the basis of geologic, geomorphic, and hydrologic factors at a particular volcano. Least-squares best fits to plots of flow-front travel time as a function of distance from source provide predictive second-degree polynomial equations with high coefficients of determination for four broad size classes of flow based on near-source flow rate: extremely large flows (>1 000 000 m3/s), very large flows (10 000–1 000 000 m3/s), large flows (1000–10 000 m3/s), and moderate flows (100–1000 m3/s). A strong nonlinear correlation that exists between initial total flow volume and flow rate for "instantaneously" generated debris flows can be used to estimate near-source flow rates in advance. Differences in geomorphic controlling factors among different flows in the data sets have relatively little effect on the strong nonlinear correlations between travel time and distance from source. Differences in flow type may be important, especially for extremely large flows, but this could not be evaluated here. At a given distance away from a volcano, travel times can vary by approximately an order of magnitude depending on flow rate. The method can provide emergency-management officials a means for estimating time windows for evacuation of communities located in hazard zones downstream from potentially hazardous volcanoes.

Fitting competitive adsorption isotherms to the experimental distribution data in reversed-phase systems

Distribution isotherms of phenol and resorcinol and o-cresol were determined between aqueous solutions with different concentrations of methanol and an octadecyl silica adsorbent. Single-component distribution data can be well described by either the Langmuir or the Jovanovic adsorption isotherms. The competitive Langmuir isotherm with single-component isotherm coefficients can describe moderately successfully the distribution in mixtures of phenol and o-cresol in 20–40% methanol, but it does not fit the experimental distribution data of the mixtures of phenol and resorcinol, which are better described by the competitive Jovanovic and the quadratic isotherms with single-component coefficients. Generally, the quality of the fit of the isotherms improves as the concentration of methanol in the solutions increases. The dependencies of the logarithms of the Langmuir isotherm coefficients and of the coefficients of quadratic and Jovanovic isotherms on the composition of the mobile phase can be described by second-degree polynomial equations over the range 0–40% methanol in water.

Blind separation of sources using a new polynomialequation

The authors present a simple method for estimating the mixing matrix in the two source separation problem. It is proven that the separation can be obtained by solving a second-degree polynomial equation that involves fourth-order cumulants.

The Effect of Composition on the Tableting Indices of Binary Powder Mixtures

ABSTRACTThe purpose of this study was to investigate the effect of the composition of a binary powder mixture on the bonding index, the brittle fracture index, and the strain index, as defined by Hiestand. The studies involved tensile strength and dynamic indentation hardness determinations of square compacts, the solid fractions of which were 0.83. The mixtures were such that both components consolidated by plastic deformation, both components consolidated by brittle fracture, or one component was brittle while the other was plastic. The measured quantities were then used to compute the bonding index, the brittle fracture index, and the strain index. The bonding indices and tensile strengths of the individual plastic materials were greater than those of the individual brittle materials. It was concluded that the bonding indices were linearly related to composition when both materials consolidated by the same mechanism. It was further concluded that the bonding indices were related to compact composition by a second-degree polynomial equation for mixtures with one brittle and one plastic component. This latter relationship was consistent for four pairs of components.

On triple focusing dipole magnets

We present a general, paraxial study of triple focusing (i.e., stigmatic and non-dispersive) index-free dipole magnets. The transcendental equations which describe such magnets lead to a second-degree polynomial equation which we solve analytically. The two solutions of this equation correspond to magnets having either one or no intermediate focal points in the vertical direction. The first-order optical properties of the physical solutions are studied.

Pressure-flow characteristics of intramural and total coronary collateral networks.

The purpose of the present investigation was to determine the pressure-flow (PF) relationship of intramural collaterals and to determine whether their characteristics differ significantly from those of the total collateral network, defined as the epicardial plus intramural collateral network. Because a significant portion of the collateral flow is diverted away from the retrograde flow measurement, we embolized the coronary vessel on which the retrograde flow was measured with spheres of various sizes until the retrograde flow was maximized and retrograde flow diversion blocked. The PF relationship was obtained before and after the epicardial collaterals were cauterized to determine the characteristics of the total and intramural collateral network. PF data for the collateral circulations were obtained by changing the inflow pressure to all coronary vessels simultaneously and by measuring the retrograde flow while maintaining the retrograde outflow pressure at 0 mmHg. The PF characteristics of the total and intramural collateral circulations could be fitted by either a second-degree polynomial or linear equation. In both cases the pressure intercept crossed the origin of the axes. The average contribution of intramural collaterals to total retrograde flow was 58 +/- 5%. We conclude that the PF characteristics of intramural collaterals parallel those of the total collateral circulation.

Ultrasound measurements of fetal limb bones.

In order to establish the growth patterns of fetal limbs, measurements of femur, humerus, tibia, fibula, radius and ulna were made by ultrasound and related to gestational age. To this end, 2317 normal singleton pregnant women were studied cross-sectionally at 13-40 weeks of gestation. Patients were selected on the basis of a certain last menstrual period, history of regular cycles and at least one ultrasound scan before 16 weeks confirming gestational age. Linear growth of all limb measurements was observed between 13 and 28 weeks of gestation. From this gestational age onwards, a flattening of the growth curve was seen. A second-degree polynomial equation turned out to be the best model to describe this phenomenon. The measurements of all six fetal long bones showed a high correlation with menstrual age (r >or= 0.99). The femur displayed the largest mean weekly increments (2.8 mm per week from 13 to 28 weeks and 1.7 mm per week from 29 to 40 weeks of gestation) and the radius had the smallest (2.08 mm per week from 13 to 28 weeks and 1.25 mm per week from 29 to 40 weeks' gestation). Considering inter and intraoperator variations and the weekly increment of fetal long bone length, a correct evaluation of limb growth is possible for the femur every week before 28 weeks and every 2 weeks after 28 weeks. For the remaining limb bones, a correct evaluation is possible every 2 weeks at all gestational ages.

Head-Capsule Width of Lobesia botrana (Lepidoptera: Tortricidae) Larvae Reared on Three Different Diets

Both sexes of Lobesia botrana (Denis & Schiffermueller) were found to have five instars, as determined by measurements of head-capsule widths of larvae reared in the laboratory on healthy grape berries, berries infected by the fungus Botrytis cinerea , or on an artificial diet. In the first two instars there were no significant differences in widths of head capsules between larvae reared on the different diets. However, differences became more pronounced in later instars and they were significant between the three diets in fifth instars. Larvae fed on infected berries generally had the largest widths. There was no overlapping in head-capsule width between the various instars. The increase in head-capsule widths approximately followed Dyar's rule but is more accurately described by a second-degree polynomial equation.

バルプロ酸単独服用患者における血清蛋白との結合

Valproic acid is highly bound to serum protein, mainly albumin. Valproic acid exhibits concentration-dependent protein binding, which partly accounts for considerable variability in the unbound fraction, even within the therapeutic range. Therefore, valproic acidprotein interaction and determination of the corresponding binding parameters should be considered. In the present study, we investigated the binding characteristics in epileptic adults and children under valproic acid chronic monotherapy, and determined the binding parameters of valproic acid to serum protein.There was a curvilinear relationship between total and unbound concentrations orunbound fraction, using a second-degree polynomial equation. The unbound fraction increased curvilinearly in proportion to the increase in total concentration . Averages of unbound fraction obtained from adults and children were not significantly different from each other (P>0.05). The unbound fraction ranged from 4.1 to 16 .7% (adults), and from 5.6 to 16.7% (children). There were marked about three-fold variations of unbound fraction in both adults and children.On the other hand, as the Scatchard plot was shown to be linear in both adults and children, valproic acid was bound to only one class of binding sites on the protein molecule. The binding parameters calculated by Scatchard plot were as follows: association constants (Ka) were 1. 14×104l/mol (adults) and 1.44×104l/mol (children), total number of binding sites (n (Pt)) was 1.36×10-3mol/l (adults) and 1. 20×10-3mol/l (children). There was no significant difference in the two parameters between adults and children (P>0 .05). Therefore, we determined the binding parameters from all samples (n=96) by using Langmuir plot. These results showed that dissociation constant (Kd) was 3 .89×10-5mol/l, total number of binding sites (n (Pt)) was 8.46×10-4 mol/l.These results suggested that valproic acid had the same binding characteristics in both adults and children. The molecular interaction between valproic acid and serum albumin involved one class of binding sites within the investigated range.

Influence of European Corn Borer (Lepidoptera: Pyralidae) Feeding on Various Stages of Field Corn in Kansas

The impact of second-generation European corn borer (ECB), Ostrinia nubilalis (Hübner) stalk tunneling on yield reductions in corn, Zea mays L., was investigated for the Kansasproduction system. The stage of corn development when larval tunneling was initiated, vertical location of larvae on the plant, and number of larvae per plant were variables in the study. The probability of ECB larval occurrence at nodes above ground level was highest on the ear node; larval frequency decreased rapidly as distance increased from the ear node. Acrossall fields, the average probability of larval occurrence per stratum was 18.4, 59.5, and 22.1% for the bottom, middle, and top strata, respectively. A significant negative linear relationship between grain yield and level of infestation in the bottom and middle strata was fitted for all plant stages, with the exception of bottom-dent in 1981 and middle-dough in 1982. No relationship was seen between grain yield and level of infestation in the top stratum regardless of plant stage. The relationship between percentage yield reduction and cumulative corn growing degree-days (GDD) remaining to corn physiological maturity was described using a second-degree polynomial equation. Maximum potential yield reduction as a result of tunneling initiated by third-instar larvae was predicted to occur when ca. 35.7% of the corn GDD remained to physiological maturity. This coincided approximately with initiation of the blister stage. The percentage yield reduction per borer decreased as tunneling was initiated nearer to physiological maturity. Larval tunneling initiated during the pollination period of corn development appeared to have a smaller effect on yield than tunneling initiated near blister stage.

The estimation of solubility in binary solvents: application of the reduced 3-suffix solubility equation to ethanol-water mixtures.

The reduced 3-suffix solubility equation (R3SSE) is applied to the characterization of solubility in the ethanol-water system. The data needed are the solubility of the compound in each of the pure solvents and at one ethanol-water composition. This composition has been estimated from solubility data to be 0.56 volume fraction of ethanol. The solubility obtained at this volume fraction is used to estimate the ternary solute-solvent interaction constant, C2. The R3SSE, with the C2 thus obtained, predicts the mixed solvent solubilities of the compounds tested, as accurately as that obtained from several volume fractions. The superiority of the R3SSE over two related equations--a simple second-degree polynomial equation and a simplified form of the R3SSE which neglects contributions to solubility from the solvent mixture--is also demonstrated for a number of solutes.

Responses of Aquatic Invertebrates to Declining Oxygen Conditions

Several models for the analysis of data relating the rate of oxygen uptake to environmental oxygen level have been evaluated. We conclude that the quadratic (or second-degree) polynomial, though hardly perfect, is the best. Data from 31 species of aquatic invertebrates are described by constants of the quadratic (or second-degree) polynomial equation. The results suggest a phylogenetic trend of increasing regulation of aerobic metabolism in response to declining environmental levels as animals acquire structures that effectively insulate their respiring tissue from the habitat. Many of these species apparently cease withdrawing oxygen from their external environment long before they have exhausted its supply. Presumably, those species with long lasting internal oxygen reservoirs, such as gas bubbles or pools of high oxygen affinity hemoglobin, continue to operate aerobic pathways, but those without substantial oxygen storage devices must switch over to anaerobic pathways, despite the availability of small residual volumes of external oxygen.

Measurement and Prediction of Viscosity-Pressure Characteristics of Liquids

The successful application of a precision capillary-type pressure viscometer in the measurement of viscosity-pressure characteristics of a variety of fluids over the range of 0 to 10,000 psig is demonstrated. A number of pressure-viscosity values determined for about ten different chemical types of fluids provides the basis for the development of a graphical relationship involving the logarithm of viscosity, the pressure coefficient, and the viscosity temperature properties (ASTM slope) of the fluid. A second-degree polynomial equation in conjunction with a digital computer program allows one to calculate the pressure coefficient and, therefore, the viscosity values of fluids at 100 deg F and pressures up to 10,000 psig. The derivation of an exponent for the pressure coefficient extends the ability to calculate values over a range of temperatures. Primary data from three sources are used to compare and contrast these methods with those proposed by several other investigators.