Polynomial Equation Of Degree Research Articles

VEGETATION TEMPORAL VARIATION ANALYSIS WITHIN "PADUREA CENAD" PROTECTED AREA BASED ON REMOTE SENSING

The study used the technique based on remote sensing to analyze the temporal variability within the "Cenad Forest" Protected Area, Timis County, Romania. The Sentinel 2 satellite system was used to obtain satellite images. Images were taken in the four seasons, the period 2020-2022. The result was a package of 12 sets of satellite data, one for each anotype and year of the study period. Based on the spectral values, the NDMI, NDVI and NBR indices were calculated (series of 908 data for each index, season and year). Descriptive statistical analysis was used for the primary evaluation of the data. In the case of 2020, the variation of the NDVI index in relation to NDMI and of the NBR index in relation to NDMI and NDVI was described by polynomial equations of degree 2, with the exception of moment t3, when NBR in relation to NDVI was described by a linear equation. In the case of the year 2021, the variation of the NBR index in relation to the NDVI (at all four moments of determination) was described by linear equations, and in the case of the other indices the variation was described by polynomial equations of the 2nd degree. In the case of 2022, the variation of the NBR-t1 index in relation to NDVI-t1 was described by a polynomial equation of the 2nd degree, and in all other cases, the variation of the indices was described by linear equations. In the case of 2022, the average values of the indices showed lower levels than the average values recorded in 2020 and 2021. The comparative analysis of the data series, for each index, from the years 2020 and 2021 compared to the year 2022, led to statistically safe differences (p<0.001). An exception was registered in the case of NDMI-t4 (year 2020) and NDVI-t1 (2021) when the differences did not present safety (p>0.05).

Non-Homogeneous Binary Cubic Equation a(x-y)<sup>3</sup>=8bxy

Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)<sup>3</sup>=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

Non-Homogeneous Binary Cubic Equation a(x-y)<sup>3</sup>=8bxy

Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)<sup>3</sup>=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

New Theorems Solving Fifth Degree Polynomial Equation in Complete Forms by Proposing New Five Roots Composed of Radical Expressions

This paper is presenting new theorems and formulas to solve fifth degree polynomial equation in general forms by proposing five formulary solutions that we can calculate nearly simultaneously. The proposed roots for fifth degree polynomial in this paper are composed of radical expressions distributed in a specific architecture that allows them to neutralize each other during multiplication by eliminating radicality and reducing degrees of terms. The proposed architecture for each solution of quantic polynomial is composed of at least six distributed terms, where each term is a multiplication of a pair of radical expressions, whereas multiplications among proposed terms is allowing the reduction of degrees in order to obtain a quartic form at maximum. Furthermore, the proposed architectures of radical roots in this paper are allowing also to solve polynomial equations with degrees higher than five by reducing their expressions. As a result, this paper is presenting two new theorems to solve quantic equation in general forms, where one theorem is eliminating the expression of fourth degree then reducing the whole fifth degree equation into a simplified form, whereas the other new theorem is solving the quantic equation without eliminating the expression of fourth degree. All proposed theorems in this paper are built on a scaling logic concretized by engineering the structure of solutions then calculating the precise formulas of roots for equations, which allow to calculate all roots nearly simultaneously and forward the used engineering methodology to solve polynomial equations with degrees higher than five.

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

릴의 방법과 지오지브라, 파이썬을 이용한 다항방정식 풀이

A major problem in algebra was to find a formula for the roots of polynomial equations of degree 5 or more, and it was proved by Abel and Ruffini that there was no formula for the roots. Lill developed the ‘graphical method’ by exploring the roots of polynomial equations using a geometric method rather than an algebraic method. To solve arbitrary polynomial equations, certain conditions of the ‘graphical method’ must be satisfied, but Lill did not provide a specific method for this. Beloch expressed quadratic and cubic equations using Lill's method and then discovered a method of satisfying specific conditions using origami. However, when the degree is 4 or more, several conditions must be satisfied simultaneously to solve it with origami, so there has been no research on solving the roots of arbitrary polynomial equations. Therefore, this study selected the following research questions. First, we develop a program that automatically solves quadratic and cubic equations using Lill's method, geogebra, and python. Second, we develop a program that automatically solves arbitrary polynomial equations of degree 4 or higher using Lill's method and python. As a result of the study, we were able to manually find the paths of quadratic and cubic equations using Geogebra and automatically find them using Python and develop a program to solve them using Lill's method. To find the path of an arbitrary nth degree polynomial equation of degree 4 or higher, we used object-oriented programming from the Pygame library to solve it. The significance of this research is that it visualizes the process of approximating the roots of an arbitrary polynomial equation, approaches it with a geometric method rather than an algebraic method, and develops a program to solve arbitrary polynomial equations in a generalized way.

Studying the Recurrent Sequence Generated by Power Function using QUATTRO-20

We presented the bifurcational diagram of power function Fi(x) = r·x·(1 – x^2) which could be treated as first approximation of trigonometric function F(x) = r·x·cos x. Using second composite Fi^2(x) in analytical form and solving 8-th degree polynomial equation bifurcational diagram with period doubling 1, 2, 4 was obtained and attractors were established. Analytical solutions of expressions x = Fi^2(x) allows us to establish the fixed point attractors and periodic attractors in interval (-V5,V5). Bifurcation diagram obtained analytically was compared with its aproximate analogue Finite State diagram. ‍

Motion and zero velocity curves of a dust grain around collinear libration points for the binary IRAS 11472-0800 and G29-38 with a triaxial star and variable masses

This paper investigates motion and zero velocity curves of a dust grain around collinear libration points (CLPs) of the circular restricted three-body problem (R3BP) when both stars IRAS 11472-0800 and G29-38 vary their masses according to the unified Mestschersky law (UML) and their motion is described by the Gylden-Mestschersky problem (GMP) with further assumption that the bigger body is a variable triaxial star. The non-autonomous equations of motion of the model are derived and transformed to the autonomized forms using the Mestschersky transformation (MT), the UML, the particular integral and solutions of the GMP, and a transformation we introduced which enable us to convert the time dependent triaxiality of the bigger star to constant triaxiality. The CLPs are examined and it is seen that when motion takes place on the plane joining the stars, there are three CLPs. To further validate the existence of the CLPs, we obtain three polynomial equations of degree seven for the three CLPs, respectively and we numerically explored the roots. It is seen that only three CLPs exists for all values of the mass parameters coupled with triaxiality of the bigger star. The stability of these points is studied and it is seen that CLPsL1 andL2 are unstable for all mass parameters throughout the interval 0<κ<∞, whereκ is a constant of the particular integral of the GMP and depicts the mass variation parameter of the stars. However, CLPL3can be stable and unstable due to the triaxiality of the bigger star, mass parameters and the mass variation parameter. Further, the zero velocity curves (ZVCs) of the dust grain around the CLPs are investigated and the effects of the system parameters divulged. It is seen that, the mass parameters and mass variation parameter can increase or decrease the region where motion of the dust grain is dynamically forbidden, while triaxiality of the bigger star increases the region where motion is forbidden around the CLPs and it allows the appearance of a petal-shaped ZVC around the bigger star. Finally, the orbits of the dust grain around the CLPs is explored, and it is seen that the paths around each collinear point differs. The deviations in the paths are due to the perturbing effects of the mass parameters, triaxiality of the bigger star and the mass variation parameter of the stars. Our problem could have important physical applications when studying celestial bodies of triaxial shape with changing masses of the bodies. In particular, we can assume that the binary system may describe a host star along with an exoplanet, or a massive exoplanet with an exomoon whose masses vary with time.

New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously

New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously

Mathematical Modeling of Split Tensile Strength of Agrostone with Different Bio-fillers

There has been a growing imbalance in the demand and supply of construction materials with increasing prices day by day at an alarming rate due to escalating cost of energy. Quest for a low cost, environmental friendly material has resulted in the development of Agrostone made out of industrial and agricultural waste while saving the environment from solid waste disposal issues. This also contributes for saving the energy required for production of conventional building materials up to 20 per cent thereby reducing the emissions. Agrostone made of different bio-fillers such as water hyacinth, bagasse and grass, satisfy the strength requirements as per Ethiopian construction material standard which is equivalent to a hollow block. The split tensile strength has been evolved through the compression test by adopting the standard test methods of applied for concrete. This paper presents the correlation of the experimental results split tensile strength with the values obtained using various empirical relations existing in the literature for unreinforced concrete. It is found that all these relations have underestimated the results while the experimental results of Agrostone with water hyacinth bio-filler have better correlation with results of relation given by Arioglu et al. This paper also presents mathematical models developed using linear and polynomial regression analysis to predict the split tensile strength of Agrostone. The correlation coefficients from the regression analysis are obtained using the Stats.blue software. It is found that the polynomial equations of degree five and the correlation indicators r and r2 show a strong relation between the independent compressive strength and the dependent split tensile strength. The linear regression also predicts the values with acceptable degree of accuracy. The resultsobtained by developed mathematical models are very close to the experimental within acceptable range of error.

Revisiting the general cubic: a simplification of Cardano's solution

Given a polynomial equation x3 + ax2 + bx + c = 0 of degree 3 with real coefficients, we may translate the variable by replacing x with x − to make the quadratic term vanish. We then obtain a simpler equation x3 + px + q = 0 where Therefore, in order to solve a polynomial equation of degree 3, it is sufficient to solve equations of the form x3 + px + q = 0.

Approximation of intra-particle reaction/diffusion effects of immobilized enzyme system following reverse Michaelis–Menten (rMM) mechanism: third degree polynomial and Akbari–Ganji methods

Two approximate analytical expressions based on third degree polynomial and Akbari–Ganji’s method (AGM) were derived for the reaction/diffusion controlled kinetics of an immobilized enzyme (IE) systems. The approximation methods predict substrate concentration profile and effectiveness factor (eta) in a porous spherical particle. The reaction is assumed to follow reverse Michaelis–Menten (rMM) kinetics. The approximate methods predictions were comparable to that of numerical solution (using the Matlab finite difference function, bvp4c) at wide range of {phi }^{2} and {y}_{o} especially at low {phi }^{2} and high {y}_{o} (polynomial equation) and low {phi }^{2} and low {y}_{o} (AGM equation). Although the approximate solution was derived for rMM kinetics, the results can be used to describe other important enzymatic reaction kinetics such as simple Michaelis–Menten (MM) kinetics and MM with competitive product inhibition kinetics. A necessary design equation should be satisfied when using polynomial or AGM approximation for different enzyme kinetic equations. In this work, two examples of enzymatic reactions of industrial interest were studied, namely glucose-fructose isomerization follows rMM kinetics and hydrolysis of lactose follows Michaelis–Menten (MM) equation with competitive product (galactose) inhibition. Predictions of the developed third degree polynomial and AGM approximation equations agree with that of numerical solution, the percentage relative error for the effectiveness factor was less than 11 in comparison with the numerical solution. Good agreement between approximate and numerical estimations demonstrates the validity of these approximation methods.

Basins of attraction of a one-parameter family of root-finding techniques

Abstract Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (A = 0), Halley’s (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley’s method.

Statistical Study of Bisection Method for Cubic Equations with Random Coefficients

The bisection method is an iterative approach used in numerical analysis to find solutions to nonlinear equations. The main purpose of this paper is to study how the parameters of a probability distribution characterizing the coefficients of a cubic polynomial can influence the convergence of the bisection method. The study covers discrete and continuous distributions, including discrete uniform, continuous uniform, and normal distributions. It was found that for both types of uniform distribution inputs, a second-degree polynomial equation can predict the average iteration for a given parameter r, where r indicates the distribution interval [−r, r]. Interestingly, the coefficients of the second-degree polynomial are nearly identical for discrete and continuous uniform distributions. For normal distribution input, the average iteration does not depend upon the standard deviation when the mean is fixed and the standard deviation is varying. But when the standard deviation is fixed and the mean is varying, the second-degree polynomial is still the best fit. This means the average iteration depends upon the mean of the normal distribution. Overall, our paper concludes that: I. For uniform distribution input, the average iteration does not depend on whether the distribution is discrete or continuous but rather depends on the range of the distribution which is its parameter. II. For non-uniform distribution input, the average iteration depends on the mean of the distribution (location parameter) but not on the standard deviation (scale parameter). Finally, a curtain is raised in the future direction of research in which we propose to combine the bisection method with the regula falsi and Newton-Raphson methods to increase the rate of convergence.

CHARACTERIZATION OF SOME AROMATIC AND MEDICINAL PLANTS SPECIES BASED ON UAV IMAGES

The study used imaging analysis to characterize several species of aromatic and medicinal plants. UAV images were taken on five aromatic and medicinal plants species, cultivated by SCDA Lovrin, Timis county, Romania. The biological material was represented by Lavandulua x intermedia (Lavandin), P1; Hyssopus officinalis L., P2, P6; Origanum vulgare L., P3; Thymus serpyllum L., P4; Salvia officinalis L., P5 (P1 to P6 species code in the article). The images were analyzed (four repetitions) and the values of the RGB color parameters were obtained. The ANOVA test confirmed the presence of variance and the safety of the data (F&gt;F crit, p&lt;0.001; Alpha=0.001). The correlation analysis highlighted positive and negative correlations of different intensity levels between RGB parameters and calculated ratios (R/G, R/B, G/R, G/B, B/R, B/G). The variation of G in relation to R was described by a polynomial equation of degree 3 (R2=0.956), and the variation of B in relation to R was described by a polynomial equation of degree 2 (R2=0.736). High variability was recorded in the case of parameter B (CVB=23.44534), plant P3 (Origanum vulgare L.), and low variability was recorded in the case of color parameter G (CVG=2.66030), plant P2 (Hyssopus officinalis L.). Based on PCA, the distribution diagram of plant species in relation to RGB color parameters and calculated ratios, as biplot, was obtained. PC1 explained 66.037% of variance, and PC2 explained 33.095% of variance in relation to RGB, and PC1 explained 58.923% of variance, and PC2 explained 36.46% of variance in relation to the calculated ratios. The cluster analysis facilitated the grouping of plant species (P1 to P6) based on similarity in relation to the RGB color parameter values (Coph.corr. = 0.959).

PRELIMINARY RESEARCH ON THE PLANT STATUS OF SEVERAL WHEAT VARIETIES IN RELATION TO NITROGEN MINERAL FERTILIZATION

The study evaluated the variation of plant height in relation to nitrogen fertilization, and took place within the Didactic and Experimental Resort (DER) of the University of Life Science "King Mihai I" from Timisoara. Six wheat varieties were studied, three with smaller height (C3 – Montecarlo, C4 – Venezio, C5 – Combin) and three with larger height (C1 – PG102, C2 – Falado, C6 – Pitar). Variable doses of nitrogen (ammonium nitrate) were applied in the range of 0-280 kg N a.s. ha-1 (a.s. – active substance). In the experimental conditions (chernozem soil) and associated with the climatic conditions, according to the genetic specificity, the wheat varieties considered in the study used the applied fertilization in a different way. The height of the plants varied between 63.67±1.45 - 73.00±1.15 cm in the C1 variety; 62.33±0.88 - 73.67±2.33 cm for the C2 variety; 56.67±1.20 - 63.67±1.45 cm for the C3 variety; 54.00±2.08 - 62.67±3.18 cm for the C4 variety; 59.00±0.58 - 70.00±1.15 cm in the C5 ace; 67.33±1.76 - 77.67±1.20 cm for the C6 variety. The data series presented a normal distribution, and a close relationship with the N doses (r=0.994 for the variety C1; r=0.980 for the variety C2; r=0.984 for the variety C3; r=0.979 for the variety C4; r=0.985 for the variety C5; r=0.983 for the C6 variety). Based on the PCA, the wheat varieties studied were positioned specifically in relation to the applied fertilization; PC1 explained 92.498% of variance, and PC2 explained 4.134% of variance. The cluster analysis facilitated the clustering of variants from the perspective of wheat varieties (Coph.corr.=0.732) and from the perspective of fertilization (Coph.corr.=0.727). A ranking of the wheat varieties was obtained based on the plant height values (Ph), in response to the applied fertilization. The variation of Ph depending on the doses of N was described by polynomial equations of degree 2 and 3, under conditions of statistical safety.

VARIATION OF SOME PHYSIOLOGICAL INDICES IN Tagetes erecta L. ACCORDING TO GROWTH SUBSTRATE

The study evaluated the influence of some growing substrates with variable content of compost from biodegradable waste on some physiological indices and growth parameters in the Tagetes erecta L. species. Different components were used to prepare the growth substrates, and depending on the weight of participation, the experimental variants resulted: V1 – garden soil 100%; V2 – compost 100%; V3 – compost 20% + peat 70% + vermiculite 10%; V4 – compost 30% + peat 60% + vermiculite 10%; V5 – compost 40% + peat 50% + vermiculite 10% (each variant in four repetitions). The biological material was represented by the Tagetes erecta L. species. The genotype "Tagetes erecta F1 Discovery Orange" was cultivated. Physiological indices and biometric parameters were evaluated, considered representative to quantify the plants response to growth substrates: plant height (Ph), shoots number (Sn), plant diameter (Pd) and shoots diameter (Sd). Several determinations were made during the vegetation period, at different times: April 22 (T1), May 22 (T2), June 18 (T3), July 21 (T4) and August 27 (T5). A strong, positive correlation was recorded between plant height (Ph) and shoot diameter (Sd), r=0.881***. Moderate correlations were recorded between plant height (Ph) and shoot number (Sn), r=0.715***, between plant height (Ph) and plant diameter (Pd), r=0.754***, between shoot number (Sn ) and shoot diameter (Sd), r=0.700***, and between plant diameter (Pd) and shoot diameter (Sd), r=0.781***. Weak correlation was recorded between shoot number (Sn) and plant diameter (Pd), r=0.669*** (*** p &lt; .001). In the PCA evaluation, PC1 explained 83.581% of variance, and PC2 explained 10.035% of variance. The cluster analysis, in relation to the important parameters considered (Ph, Pd) facilitated obtaining a dendrogram of association of the variants (Coph.corr.=0.881). The dynamics of the plants main parameters (Pd, Ph) on the experimental variants in relation to time, was described by polynomial equations of the 2nd degree under differentiated conditions of statistical certainty. A variable increase (D) in the evaluated parameters (positive and negative values) was recorded in the experimental variants (V2 – V5) in relation to the control variant (V1) and the moments of determination during the study period.

On the Optimal Input Rate in Queues with Batch Service

In recent years, queuing systems with batch service are emerging as powerful and flexible mathematical models in different frameworks. In this paper, we consider a single server queuing system with Poissonian arrivals, infinite buffers, and a constant batch size b. This paper addresses a little-studied optimization problem, namely the existence of an optimal arrival rate that minimizes the average sojourn time. Unlike the classical M/M/1 queue, for any batch size b, the problem admits a non-trivial solution that can be found by solving a polynomial equation of degree b+1. Since, in general, only numerical solutions are available, a simple first-order approximation is also derived and the corresponding deviations (in terms of input rate and sojourn time) are calculated. In more detail, it is shown that the approximation improves as the batch size increases and, in any case, the relative error for the average sojourn time is less than 0.34%. Finally, the paper provides new theoretical results about the asymptotic service rate in the equivalent birth–death process, highlighting how it depends on all queue parameters.

Kinematics of the Hybrid 6-Axis (H6A) manipulator

AbstractThis paper presents a comprehensive study of the forward and inverse kinematics of a six-degrees-of-freedom (DoF) spatial manipulator with a novel architecture. Developed by Systemantics India Pvt. Ltd., Bangalore, and designated as the H6A (i.e., Hybrid 6-Axis), this manipulator consists of two arm-like branches, which are attached to a rigid waist at the proximal end and are coupled together via a wrist assembly at the other. Kinematics of the manipulator is challenging due to the presence of two multi-DoF passive joints: a spherical joint in the right arm and a universal in the left. The forward kinematic problem has eight solutions, which are derived analytically in the closed form. The inverse kinematic problem leads to $160$ solutions and involves the derivation of a $40$ -degree polynomial equation, whose coefficients are obtained as closed-form symbolic expressions of the pose parameters of the end-effector, thus ensuring the generality of the results over all possible inputs. Furthermore, the analyses performed lead naturally to the conditions for various singularities involved, including certain non-trivial architecture singularities. The results are illustrated via numerical examples which are validated extensively.

ИЗМЕНЕНИЕ ТЕРМИЧЕСКИХ РЕСУРСОВ ВЕГЕТАЦИОННОГО ПЕРИОДА И УРОЖАЙНОСТЬ ЯРОВОЙ ПШЕНИЦЫ В УСЛОВИЯХ СРЕДНЕГО ПОВОЛЖЬЯ

The article presents the results of long-term studies (1982-2018) on the assessment of changes in thermal resources during the growing season and the indication of their impact on the yield of spring wheat. The work was carried out in the conditions of the northern part of the Middle Volga region on gray forest soils. The soil of the experimental site is gray forest medium loamy, the content of humus (according to Tyurin) is 2.8...3.2%, mobile phosphorus and potassium (according to Kirsanov) - 105...184 and 79...149 mg /100 g of soil, respectively, pH- 5.6...5.8. During the observation period, the highest temperature increase was recorded in July (2.4°C) and May (2.2°C), and the lowest in June (1.0°C). The sum of effective temperatures over 10°C increased by 500°C during the study period. The yield of spring wheat grown without fertilizers varied greatly in different years, ranging from 0.99 to 2.97 t/ha. The highest value of this indicator was observed at an average monthly air temperature of about 13 0C in May, 17 0C in June, 18 0c in July, and the sum of active temperatures was 1800 0c. The studied thermal resources, in accordance with the significance of the impact on the yield of spring wheat, can be arranged in the following descending order: average air temperature in June &gt; average air temperature in July &gt; the sum of effective temperatures above 10 °C &gt; average air temperature in May. To describe the dependence of the yield of spring wheat grain on thermal resources, in three cases out of four, a polynomial equation of the second degree was better suited, and in one case, an exponential equation.