A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].

A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].

Correlations among Thermal Conductivity, Porosity, and Permeability of Mesozoic Sandstones and Siltstones from the Southern Regions of the West Siberian Plate

—We consider results of measurements of thermal conductivity, porosity, and permeability for 780 samples of Mesozoic sandstones and siltstones from the cores of 50 wells drilled in three southern regions of the West Siberian Plate (Novosibirsk and Tomsk regions, Surgut region of the Khanty-Mansi Autonomous Area). The thermal conductivity of the samples was measured twice: in dry and in water-saturated states. It has been established that the thermal conductivity of water-saturated rocks is on average 20–40% higher. The thermal conductivities of dry and water-saturated samples show stable correlations between each other and with the sample porosity and permeability. These correlations can be used for the approximate estimation of the thermal conductivity of water-saturated rocks from the measured thermal conductivity of dry samples or even from the porosity values. The relationship between thermal conductivity and porosity can be used for the rapid assessment of rock porosity from the measured thermal conductivity of the core.

Approximate estimation of the critical diameter in Koenen tests

A simple correlation between computed detonation parameters and the critical diameter obtained using the Koenen (steel-sleeve) test is reported. This correlation is not meant to replace a proper Koenen test, but rather, to act as an aid to help to know which orifice diameter to start testing with.

Decoding of EEG Signals Shows No Evidence of a Neural Signature for Subitizing in Sequential Numerosity

Numerosity perception is largely governed by two mechanisms. The first so-called subitizing system allows one to enumerate a small number of items (up to three or four) without error. The second system allows only an approximate estimation of larger numerosities. Here, we investigate the neural bases of the two systems using sequentially presented numerosity. Sequential numerosity (i.e., the number of events presented over time) starts as a subitizable set but may eventually transition into a larger numerosity in the approximate estimation range, thus offering a unique opportunity to investigate the neural signature of that transition point, or subitizing boundary. If sequential numerosity is encoded by two distinct perceptual mechanisms (i.e., for subitizing and approximate estimation), neural representations of the sequentially presented items crossing the subitizing boundary should be sharply distinguishable. In contrast, if sequential numerosity is encoded by a single perceptual mechanism for all numerosities and subitizing is achieved through an external postperceptual mechanism, no such differences in the neural representations should indicate the subitizing boundary. Using the high temporal resolution of the EEG technique incorporating a multivariate decoding analysis, we found results consistent with the latter hypothesis: No sharp representational distinctions were observed between items across the subitizing boundary, which is in contrast with the behavioral pattern of subitizing. The results support a single perceptual mechanism encoding sequential numerosities, whereas subitizing may be supported by a postperceptual attentional mechanism operating at a later processing stage.

Estimation of the tooth gear service life by wear under misalignment conditions

A method of approximate estimation of the life of a tooth gear by the wear of the teeth of gear wheels in the presence of a misalignment is proposed. The dependences of the of the tooth gear life on loading, formulas for determining the maximum wear and contact pressure are obtained. Keywords tooth gear, wear, life, engagement, misalignment, wear resistance. [email protected]