Estimation of AR (2) Model with Dependent Errors for Unbounded Stationary and Nonstationary Time Series

In this paper the GLS and the ML estimators, the variance-covariance matrix, the unbiased for the GLS and the ML estimators of parameters of AR (2) model with constant in case of dependent errors have been derived, the simulation results shown that the values of MSE and Thiel’s U in case of unbounded stationary time series for all sample size T are less than the values of MSE and Thiel’s U in case of unbounded nonstationary time series which approved that the results for unbounded stationary times series are better than the results for unbounded nonstationary times series, and the simulation results for unbounded nonstationary time series shown that by using the measurement of MSE the best case among of all cases of nonstationary which gives the smallest values of MSE is case four when the first and the second conditions of stationary conditions for AR (2) model are exists, while by using the measurement of Thiel’s U the best case among of all cases of nonstationary which gives the smallest values of Thiel’s U is case six when the second and the third conditions of stationary conditions for AR (2) model are exists.

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong S—stationary conditions from Fritz John stationary conditions. Further, we establish strong S—stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results.

SPECIFIC FEATURES OF DETERMINING THE DESIGN OF OXYGEN TUBE TIPS UNDER NON—STATIONARY CONDITIONS OF CONVERTER MELT WITH PRELIMINARY HEATING OF SCRAP

The results of calculating the parameters of the tips of oxygen lances, taking into account the preheating of scrap metal in the converter due to the combustion of solid fuels in the modern raw material conditions of a metallurgical enterprise of Ukraine, are presented. The substantiation of the design of blowing devices is provided. The results of calculating the geometric parameters and the coefficient of off-design for standard and experimental tips of oxygen lances are presented. Using numerical methods, the value of the diameters and length of the supercritical part of the nozzle was obtained. It is shown that with a decrease in the blast flow rate, it is advisable to decrease the length of the supercritical part of the nozzle, which is fully correlated with the literature data. The change in the coefficient of off-design was determined for the studied range of oxygen consumption (600—1100 m3 / min) for standard and experimental handpieces. It is shown that the value of the coefficient of off-design is in the range of 0.52—0.98 and 1.30—2.30, respectively, for 5– and 6–nozzle standard tips. For experimental tips, this figure is 1.00—1.70. Tuyeres with experimental tips, at the pressure available in the shop, reliably operate in the range of oxygen consumption 650—950 m3 / min and allow increasing the oxygen consumption at the beginning of the blowdown to 1000—1100 m3 / min and more, if necessary. The resulting saving effect from the introduction of the modernized tips is: a decrease in the oxidation of the final slag 1.22—1.70 %; reduction in the specific consumption of lime 0.29—1.81 kg / t; reduction of specific consumption of metal charge 0.26—0.38 kg/t.

Adsorption of the Petrochemical Pollutants Released at the Small Vehicle-Service Facilities on the Coal Refinery Sludge/Pyrocarbon Compositions

It has been proven that the use of coal-like adsorbents with the mosaic and hydrophobic surface structure is efficient in cleaning thewastewater from a wide range of pollutants under the condition of their uncontrolled release from various small enterprises. Thisrange includes such environmentally dangerous agents as petrochemicals and other pollutants formed at car filling, carwash stationsor other similar facilities. Technical pyrocarbon and the coal refinery sludges are readily available waste materials with high porosity,which exhibit some adsorption activity and can be utilized in water/wastewater treatment solutions. Then the adsorbents can be eitherdisposed of at the landfill areas or incinerated as components of some secondary fuels. The highest adsorption performance is achievedfor the sorbent mixture of the refinery sludges and technical pyrocarbon with the ratio of the components 4:1.If the wastewater is flowing through this composition, the degree of petrochemicals removal reaches 75-80% for the mixture sludge/technical pyrocarbon, while the pure pyrocarbon ensures the removal degree of 15-20% only. Though adsorption efficiency under stationary conditions (keeping the adsorbent and the wastewater in contact inside some decontamination pond/vessel for at least 24 h) ishigher, this option is hard-to-realize for a small car service/wash station. As an alternative, comparatively small wastewater cleaningcartridges filled with the 4:1 mixture of coal sludge and technical pyrocarbon can be recommended for preliminary decontaminationof the wastewater formed at such enterprises before their discharge to the local municipal sewerage lines.

Mathematical modeling nonstationarity of biotechnological production of lactic acid: stability

The article presents the calculated ratios of indicators determining the stationary states of the lactic acid production process. Three technologies that are most often mentioned in scientific publications are identified: the technology of using strains of microorganisms to produce biomass is a technology that is extremely rarely used; the fairly common technology of using strains of microorganisms to produce lactic acid with the consumption of the main substrate (most often glucose); the promising technology of obtaining lactic acid using, in addition to the main substrate, a component that reproduces the main substrate in the synthesis process. For each technology, the equations of material balance for stationary and non-stationary conditions, a generalized differential equation for non-stationary conditions, and a characteristic equation are given. The formulas for estimating the coefficients of differential equations and the coefficients of the characteristic equation are also given. The equations for non-stationary conditions according to the last two technologies are based on the use of the Taylor series expansion of functions with the preservation of only the first terms of the expansion, i. e. deviations from stationarity in small. The characteristic equation is formed using the eigenvalues . The methodology for all three technologies is given, which allows us to assess the stability of the considered stationary state – the Hurwitz method. For all three technologies, numerical results are obtained for estimating the coefficients of the characteristic equations Pi. Tabular values of the coefficients are given, according to which stability estimates for the dilution rate of 0.1 h–1, 0.2 h–1, 0.3 h–1 are obtained using determinants according to the Hurwitz matrix. The results of numerical estimates for the stability of stationary states for all three technologies are presented. The estimates were based on the indicators of constants published in scientific studies.

A Note on the Gravitoelectromagnetic Analogy

We discuss the linear gravitoelectromagnetic approach used to solve Einstein’s equations in the weak-field and slow-motion approximation, which is a powerful tool to explain, by analogy with electromagnetism, several gravitational effects in the solar system, where the approximation holds true. In particular, we discuss the analogy, according to which Einstein’s equations can be written as Maxwell-like equations, and focus on the definition of the gravitoelectromagnetic fields in non-stationary conditions. Furthermore, we examine to what extent, starting from a given solution of Einstein’s equations, gravitoelectromagnetic fields can be used to describe the motion of test particles using a Lorentz-like force equation.

Rotating Machinery Diagnosing in Non-Stationary Conditions with Empirical Mode Decomposition-Based Wavelet Leaders Multifractal Spectra

Diagnosing the condition of rotating machines by non-invasive methods is based on the analysis of dynamic signals from sensors mounted on the machine—such as vibration, velocity, or acceleration sensors; torque meters; force sensors; pressure sensors; etc. The article presents a new method combining the empirical mode decomposition algorithm with wavelet leader multifractal formalism applied to diagnosing damages of rotating machines in non-stationary conditions. The development of damage causes an increase in the level of multifractality of the signal. The multifractal spectrum obtained as a result of the algorithm changes its shape. Diagnosis is based on the classification of the features of this spectrum. The method is effective in relation to faults causing impulse responses in the dynamic signal registered by the sensors. The method has been illustrated with examples of vibration signals of rotating machines recorded on a laboratory stand, as well as on real objects.

Description of Acid Battery Operating Parameters

In this paper, the operating principles of the acid battery and its features are discussed. The results of voltage tests containing the measurements conducted at the terminals of a loaded battery under constant load conditions, and dependent on time, are presented. The article depicts the principles of the development of electric models of acid batteries and their various descriptions. The principles for processing the results for the purpose of the determination and description of the battery model are characterized. The characteristics under stationary and non-stationary conditions are specified using glued functions and linear combinations of exponential functions, and the electrical parameters of the battery are determined as the components of the circuit, i.e., its electromotive force, resistance, and capacity. The dynamic characteristic of the battery in the form of transmittance was determined, using the Laplace transform. Possible uses of the crankshaft driving signals as diagnostic signals of the battery, electric starter, and internal combustion engine are also indicated.

The Einstein field equation

Various aspects of the Einstein field equation are presented. First the field equation is obtained by arguing that it is the simplest equation that respects the fundamental geometric insight into gravity. Then we consider whether the equation is stable, and introduce the weak energy and dominant energy conditions. The connection between inertial motion and the distant universe (Mach’s principle) is discussed. The equation of motion of matter is obtained from the field equation, and a comparison made with electromagnetic field theory. The energy and momentum of gravitational fields in stationary conditions is discussed, and the Komar energy obtained.