The article discusses the general scientific problem of the inadequacy of mathematical models of measurement objects for two important cases: solving measurement problems using the method of joint measurements and the method of indirect measurements. In the first part of the article, the latent synonymy of the terms measurement and calculation is discussed as the primary cause of the discrepancy between the names and definitions of the terms direct measurement, indirect measurement, aggregate measurement, and joint measurement . In the second part of the article, the definitional uncertainty of mathematical models is analyzed as a result of incomplete descriptions. Both of these phenomena are key to the problem of inadequacy. The first part of the article is devoted to an analytical review of works in which the problems of calibration of the cosmological distances scale by the method of joint measurements are solved using data on photometric distances of supernovae of type SN I a.

For one-dimensional isotropic models of the Friedmann-Robertson-Walker type of the 3rd order, in 1998–1999, due to rank inversions of the redshift, perceived as a “cosmic push”, a conclusion was made on the “acceleration of the expansion of the Universe” based on regression analysis. Based on the same data on photometric distances in 2016, taking into account the right ascension and declination of SN Ia for a three-dimensional anisotropic model of the 2nd order, a more accurate result was obtained using the cross-observation error inadequacy scheme (maximum compactness method). At the same time, it has been established that the causes of rank inversions are not the super-massive large-scale inhomogeneities of the universe, attractors, but gravitational dipoles pairs of giant super clusters of galaxies and giant voids in opposite directions of the celestial sphere. In such dipoles, the gravitational effect on objects between them is not balanced and manifests itself as an additional repulsive force from the giant voids. Five such dipoles have been discovered, and two of them have been independently confirmed. Thus, the alternative hypothesis that the Milky Way is moving towards a giant system of super clusters of galaxies at the northern galactic pole, leaving behind a giant system of voids in the constellations of Eridanus, Cetus, and Aquarius at the southern galactic pole, is more plausible according to the criterion of minimizing the error of inadequacy.

The parameters of the Earth's rotation determine the relationship between the terrestrial and celestial reference systems. In the Russian Federation, the task of determining and predicting the parameters of the Earth's rotation is assigned to the State Service of Time, Frequency and Determination of the Parameters of the Earth's Rotation. When predicting the parameters of the Earth's rotation, such modes of free rotation of the Earth as the Chandler pole motion and the free nutation of the core have a special influence. These modes are also of great theoretical importance, since their frequencies are part of the transfer function of the accepted theory of nutation and precession of the Earth MAS2000. The measurement data of the pole motion and the displacement of the celestial pole obtained by the International Earth Rotation and Reference Systems Service (IERS) have been preprocessed. The processing was carried out in order to exclude trends from these data, as well as the annual movement of the pole (for pole coordinates). Lomb-Scargle periodograms of the preprocessed measurement data are constructed. A resonant model of the Chandler pole motion based on solving the equations of free oscillations is proposed and substantiated. To verify it, we used a comparison of two Lomb-Scargle periodograms, one of which is calculated using an idealized resonance model of the Chandler pole motion, and the second is based on the results of preprocessed measurement data. The analysis of these periodograms made it possible to specify the period of the Chandler pole motion. The value of the period of the Chandler pole motion is equal to (432.47±0.03) days relative to the rotating Earth. Also, the coincidence of the Lomba-Scargle periodograms of the idealized model and the model based on the results of preprocessed measurement data confirmed the possibility of using the same principle to build a more realistic model. A numerical implementation of a more realistic resonant model of the Chandler pole motion is constructed. The parameters of this model and the results of comparing the calculated contribution to the pole coordinates with the preprocessed measurement data are presented in this article. The behavior of the Lomb-Scargle periodogram, constructed from preprocessed measurement data, was explained using the resonant model of the Chandler pole motion without involving additional assumptions about frequency splitting or slow changes in the resonant properties of planet Earth over time. The constructed (more realistic) resonance model can be used to predict the contribution of the Chandler pole motion to the pole coordinates, which leads to increased forecasting accuracy and theoretically confirms the single-frequency structure of the free nutation of the nucleus, adopted in the current theory of precession and nutation. The impossibility of using a more realistic resonance model to predict the contribution of free nuclear nutation to the coordinates of the celestial pole shift from a theoretical point of view confirms the contradiction between the single-frequency structure of free nuclear nutation accepted in the current theory of precession and nutation and the lack of confirmation of this by measurement results.

Nonparametric estimates of the probability density of a random variable are considered, for example, those used in pattern recognition and automatic classification algorithms with a priori uncertainty. With small and limited sample sizes of random variables, the variance of the traditional nonparametric probability density estimation increases. Under these conditions, it is recommended to use an integral estimate of the probability density to reduce the variance. To evaluate the areas of competence of the integral probability density estimation and nonparametric Rosenblatt-Parsen statistics, their approximation properties are compared.

The method of synthesizing an integral estimate of the probability density of a random variable is based on smoothing the traditional kernel estimate by using an integral operator. The peculiarity of the integral probability density estimate is the implicitly defined kernel function, the form of which depends on its blur coefficient and smoothing parameter. When restoring the probability density under specific statistical data conditions, the properties of the integral estimate of the probability density, in contrast to the traditional nonparametric estimate, are determined not only by the blur coefficient of the kernel function, but also by its type. To justify the effectiveness of the integral probability density estimate, the conditions of its asymptotic convergence are studied, which are compared with the traditional Rosenblatt–Parzen kernel estimate. It is found that the asymptotic values of the mean square deviation and the variance of the integral probability density estimate are less than those of the traditional nonparametric estimate. A procedure has been developed for consistently selecting the optimal blur coefficient for the integral estimation of the probability density and the smoothing parameter. The procedure for sequentially selecting the optimal blur coefficient of the integral probability density estimate and the smoothing parameter is considered. The proposed method for synthesizing an integral estimate of the probability density of a random variable can be applied in conditions of multidimensional random variables. Under these conditions, the integral probability density estimation has an advantage over the traditional nonparametric Rosenblatt-Parsen statistics. The results obtained can be used in the synthesis of pattern recognition and automatic classification algorithms in conditions of limited and small samples of statistical data.

The article considers the normalization and correction of the error of analog-to-digital converters taking into account the integral nonlinearity of their conversion function. Integral nonlinearity is a key parameter limiting the accuracy and quality of analog-todigital conversion in modern applications. Linear normalization of the error of an analog-to-digital converter, taking into account only additive and multiplicative components, is insufficient for high-precision systems, where integral nonlinearity plays a significant role.

Currently, there is no generally accepted algorithm for correcting the error of the integral nonlinearity of analog-to-digital converters. Also, there is no unified conversion function that would include linear conversion as a special case and simultaneously take into account the integral nonlinearity. For the normalization and correction of the limiting error band of analog-to-digital converters, it is proposed to use a projective measurement model. It is shown that the integral nonlinearity can be approximated by the nonlinearity of the projective conversion. A distinctive feature of the projective model is its ability to directly account for nonlinearities in conversion characteristics, eliminating their linearization or consideration as sources of error. Unlike the normalization formula in the linear model, the error bandwidth normalization formula corresponding to the projective measurement model accounts for integral nonlinearity as a separate term, hence its terminology as extended. Using examples of the characteristics of real analog-to-digital converters, the advantage of using the projective model over the traditional linear model is demonstrated. The projective model is particularly relevant for correcting the systematic measurement error bandwidth based on the generalized equation of projective measurement conversion, which includes the linear measurement equation as a special case. The results obtained are useful for metrologists, developers of analog-to-digital converters, and measurement systems. 

For the development and verification of algorithms for calculating geometric errors of inspected shafts using machine vision data, reference data is required on the coordinates of the visible boundaries of the shaft in the image, with known parameters of form deviations and positional deviations of the shaft journals, generated during the image acquisition process. To solve this problem, a method for simulating the imaging process of a rotating shaft is proposed.

A mathematical model has been developed that takes into account both the ideal shaft geometry and its actual deviations (runout, ovality, waviness, roughness), as well as the positioning errors of the shaft relative to a digital camera, which converts the optical image into an array of digital data for subsequent processing. The simulation includes generating the shaft profile, projecting three-dimensional coordinates onto a two-dimensional image considering camera parameters, and determining the coordinates of the visible shaft boundaries during its rotation. Computational experiments were performed for a shaft with a diameter of 10 mm, ovality of 0.1 mm, and a ten-point profile roughness height ( Rz ) of 50 μ m. The influence of the tilt angles of the shaft rotation axis on the image of its boundaries was analyzed. The results of calculating the coordinates of the visible boundaries of a rotating shaft during simulation of the imaging process show that the proposed method makes it possible to generate realistic data for testing algorithms, evaluate the influence of various errors, and optimize the parameters of the measurement system. It is shown that simulation is an effective tool for debugging and validating non contact methods for shaft geometry inspection. The proposed simulation method can be useful for developers of machine vision algorithms and methods for non contact inspection of shaft geometry when debugging vision systems without the need for physical reference standards, as well as for specialists in technical measurements and quality control in mechanical engineering engaged in automating shaft geometry inspection (e.g., in automotive, machine tool, and aircraft engine manufacturing). 

Effective operation of geostationary spacecraft implementing a wide range of space measurement services is contingent upon the high accuracy of their onboard attitude control system. Currently, the primary sensors for geostationary spacecraft attitude parameter determination are sun sensors, star trackers, and magnetometers. However, factors such as the high cost of sun and star sensors, their limited field of view (which hinders attitude determination during spacecraft rotation), and the high computational cost of processing star tracker data have spurred increased research into magnetometer-only attitude estimation methods capable of functioning even during highly dynamic rotation.

Nevertheless, this introduces the challenge of achieving stable and accurate estimation amidst inevitable magnetic field measurement noise caused by various random physical factors. To date, various methods and algorithms have been proposed to mitigate magnetometer noise, with the Extended Kalman Filter being among the most effective. Major drawbacks of these methods include their reliance on specific cases of spacecraft angular motion and simplified noise models, necessitating additional Extended Kalman Filter tuning tailored to specific application scenarios. In this regard, this article proposes a new approach for synthesizing a stochastic attitude estimation algorithm for geostationary spacecrafts. The algorithm is invariant to the nature of angular motion and ensures stability and required estimation accuracy under general assumptions regarding magnetometer measurement noise. The proposed stochastic filtering algorithm incorporates an additional filtering loop that utilizes the Extended Kalman Filter output estimate vector as a linear observer for the angular parameter vector. This dual-loop filtering approach ensures both stability and estimation accuracy of angular parameters, fully meeting modern requirements for geostationary spacecraft onboard attitude systems. Numerical experiment results are presented to illustrate the effectiveness of the proposed approach. 

Fiber-optic sensors operating on Mandelstam-Brillouin backscattering represent a promising class of distributed optical sensors for simultaneous temperature and deformation monitoring in extended infrastructure objects. To date the key sensor metrological characteristics have been analyzed without establishing correlations between them. However, a complex analysis of the main metrological characteristics is required when designing the sensors under discussion. A comprehensive analysis makes it possible to more accurately predict the actual metrological characteristics of a sensor under the simultaneous influence of several factors. The relevance is dictated by the need to move from estimated measurements to system design in order to ensure the required accuracy and stability of the sensor in operating conditions. A complex analysis of the sensor key functional characteristics is conducted including spatial resolution, dynamic range, measurement errors, and measurement speed. It is found that the spatial resolution of the investigated fiber-optic sensors depends on the acoustic phonons relaxation time and is limited to the value of approximately 1 m, however, the application of differential pulse methods and synthetic spectrum analysis makes it possible to obtain a resolution of 0.1–0.4 m. The sensitivity of fiber-optic sensors to the temperature changes and mechanical deformation is analyzed. The linear dependences of the Brillouin frequency shift on deformation are determined. However, the mutual sensitivity of the parameters requires the use of hybrid methods for their separation. The methods of optimizing the signal-to-noise ratio by using fiber amplifiers, single photon detectors and algorithmic filters are investigated. It is shown that the use of combined Raman amplification allows extending the measurement range up to 120 km while maintaining a spatial resolution of 5 m and a temperature error within ±(1–2) °C. The compromise relations between the key metrological characteristics of fiber-optic sensors operating on the Mandelstam-Brillouin backscattering are investigated in details. The data of the metrological characteristics analysis are critically important for defining the main trends in the sensor developments. The results obtained are of practical importance for designing the fiber-optic sensors of this type which are applied in structural monitoring the complex engineering infrastructures, including bridges, dams, tunnels. 

The quality control of milk and dairy products using innovative diagnostic methods is considered. The relevance of using methods and means of non-destructive/contactless rapid diagnostics for quality control of milk and dairy products, for example, based on the photoluminescence effect, is shown. The optimal operating modes of the luminescent laboratory bench developed earlier by the authors of this article for rapid determination of optical properties of milk have been selected. The stand allows measuring the photoluminescence of milk in the spectral ranges of protein, fat, and vitamin radiation when they are excited by LEDs with a central wavelength of 320, 362, and 445 nm, respectively. Luminescence is detected using photodiodes in the visible spectral range of 420–675 nm, 440–620 nm, and 480–660 nm, respectively. The optimal value of the pulse durations of LEDs, which is approximately 19 microseconds, has been experimentally determined, at which the highest photovoltaic luminescence is obtained. The photodiode switching delay value of about 1 microsecond was also obtained. When averaging the signal over 7000 excitation and registration cycles, the measurement time was about 15 seconds, with a relative error of no more than 3 %.

The results obtained can be used in the study of integral photoluminescence fluxes of milk, the construction of dependencies of integral photoluminescence fluxes on the content of protein, fat and other parameters in milk, for example, when conducting studies of the pH level or contamination in milk. The results obtained can provide the basis for the development of devices for non-destructive photoluminescent control of milk, which can be used on farms or in laboratories. 

The development of polymer laser carbonization methods opens up broad opportunities for creating flexible electronics components. However, a key problem remains the lack of precise data on the technological regimes of laser pyrolysis; in particular, the optimal temperature has not been established, making it impossible to ensure high quality of the synthesized materials. This paper investigates the temperature dependence of 100and 200μ m-thick polyimide films within the laser irradiation zone (wavelength 1064 nm) on laser parameters (power and pulse repetition rate) and scanning modes (beam velocity and hatch spacing) in continuous and pulsed regimes. The temperature was determined by spectral pyrometry: thermal radiation spectra were recorded using a compact HR2000+ (Ocean Optics, USA) visible-range spectrometer (350–760 nm) and transmitted to a computer for processing via specialized “Spectral Pyrometry” software.

The transmittances of the pristine films and those processed by laser scanning were determined using a spectrometric method and by measuring the transmitted power. It was found that the transmittance of the pristine 100and 200-μm-thick polyimide films at the operating laser wavelength is 87.0 % and 85.5 %, respectively. It is shown that increasing the scan line density (hatch spacing) increases the temperature in the processing zone and decreases the transmittance of the treated films. During laser processing of 100and 200μ m-thick films in the pulsed scanning mode with a hatch spacing of 20 lines/mm, their transmittance decreases to 7.5 % and 3.0 %, respectively. It was established that a heat-affected zone is formed along the laser scanning line, in which changes in the surface microstructure and optical properties of the polyimide are observed. The width of this zone depends on the laser scanning modes. The results of this work will be useful for selecting optimal laser exposure regimes in tasks involving the controlled synthesis of carbon structures, such as laser-induced graphene. 

For the purpose of disinfection of liquid media, various methods are used, for example, thermal or ultraviolet exposure, ozonation, chlorination, etc. However, these methods often turn out to be ineffective or significantly energyintensive. Heat treatment using microwave radiation has certain advantages over the above-mentioned methods of inactivation of microflora, while observing the necessary thermal treatment regime, which allows you to control the temperature parameters of the liquid. The effectiveness of microwaves is determined not only by reaching the required temperatures of the treated liquid, but also by the optimal distribution of the temperature field inside the liquid medium.

Two heat treatment modes have been studied: stationary, when the liquid inside the electrodynamic system is stationary, and non-stationary (flow-through), when the liquid is continuously moving. Three distributions of liquid temperature over the length of the installation are obtained: two theoretical dependences for stationary and flow modes and an experimental one measured in stationary mode. A method for calculating the temperature distribution of a heated flowing liquid medium based on an electrodynamic model of a microwave installation in the form of a loaded long line is proposed. In experiments on heat treatment of liquids, a traveling wave type microwave installation with transverse interaction was used, which was previously developed by the authors for processing solid materials. The practical implementation of measuring the temperature of a liquid that has been heat-treated in a microwave installation in a stationary mode is described in detail, and the technical limitations of measuring temperature under conditions of intense electromagnetic radiation are taken into account. When comparing the calculated and measured dependences of the temperature distribution over the length of the electrodynamic system of a microwave installation, the discrepancy between the calculated and experimental data was no more than 7 %. It is pointed out that there is a sharp difference in the theoretical temperature distributions for the two modes of heat treatment of a liquid medium, which must be taken into account when solving specific tasks of technological microwave processing. The results obtained are of practical interest in the development of devices for microwave disinfection of liquid media in the food, microbiological, pharmaceutical, medical, and wastewater treatment industries.

Active elements of polar cuts of ionic dielectric crystals of polar and polar-neutral classes with applied on their surface conductive coatings are used to create modern devices of piezo-, acousto-, optoelectronics operating in a wide temperature range. Conductive coatings are applied to the surface of polar slices, which leads to degradation of the crystal elements up to their destruction. However, the nature of such degradation has not yet been established. The mechanisms of electromotive force generation and ion transport in the «ionic dielectric-metal conductive coating» system have been theoretically substantiated. The main factor in the degradation of active elements is the electrochemical decomposition of polar cuts in contact with a metal conductive coating, which is caused by the difference in properties between the opposite sides of the polar cuts. Upon contact of the surface of the polar cuts of crystals, even with the same conductive coating, different products of electrochemical reactions or the same products of different concentrations are formed. This leads to a potential difference and an electromotive force, which induces short-circuit currents. The determination of the presence of short-circuit currents, their values and temperature dependences makes it possible to estimate the intensity of near-electrode electrochemical processes: the lower the strength of short-circuit currents, the weaker the electrochemical processes. This paper summarizes the results of long-term studies of short-circuit currents in a number of crystals obtained by the authors of this article. Measurements of electric currents of model lithium iodate crystals on the previously created SKIP test complex are described within the framework of the author's measurement methodology using a standard sample of high-resistance (100 mOhm). The main unit of the complex is a thermal chamber in the form of a tubular electric furnace. The successful application of the method of measuring the temperature dependences of short-circuit currents for the selection of conductive coatings of polar cuts of functional langatate crystals grown in different atmospheres with different conductive coatings is shown. It was found that as the temperature in the thermal chamber increases, the values and direction of the short-circuit currents may change due to the intensification of phase formation processes, which is confirmed by the results of phase analysis. The material of the conductive coatings that exhibited the lowest short-circuit currents and the fewest changes in current polarity was iridium.