Sum Of Squares Research Articles

Sums of squares, Hankel index and almost real rank

Sums of squares, Hankel index and almost real rank

An instance of negative variance-like ‘components’ in optics by way of applying multivariable calculus to a statistical problem

In the field of optics, the design of a lens often involves analyses of the variability in an optical parameter (such as lens focal length) under manufacturing conditions, which is done in order to identify sources of high variability that may cause the optical parameter to be out of its specification. In this interdisciplinary paper, the author shares his experience in lens design, where analyses of variability are often performed through examining the partial derivatives in a multivariable Taylor series expansion of an optical parameter to second order when interactions are expected. It is shown algebraically that when the second-order expansion is applied to express a sum of squares, negative terms in the sum can occur. This instance of negative-valued “variance-like components” is interesting pedagogically in applied mathematics and statistics, as well as from an engineering standpoint, as it suggests that, based on theory, the variance of a function of random variables may not necessarily be comprised of a simple sum of positive-valued squared deviations comprising its total variance.

Determination of children's nutritional status with machine learning classification analysis approach

<span>Malnutrition is a problem that is often faced by every country around the world. Various facts show that malnutrition is of particular concern to many researchers. To can overcome this problem, every effort has been made such as developing analytical models in identification, classification, and prediction. This study aims to determine the nutritional status of children using the machine learning (ML) classification analysis approach. The methods used in the ML analysis process consist of cluster K-Means, artificial neural network (ANN), sum square error (SSE), pearson correlation (PC), and decision tree (DT). The dataset for this study uses data on child nutrition cases that occurred in the previous and was sourced from the provincial general hospital (RSUP) M. Djamil, Padang, West Sumatera. Based on the research presented, ML performance in the nutritional status classification analysis gave maximum results. These results are reported based on the level of precision with an accuracy of 99.23%. The results of the analysis can also present a knowledge-based nutritional status classification. This research can contribute to and update the analytical model in determining nutritional status. The results of this study can also provide benefits in handling nutritional status problems that occur in children.</span>

Piecewise SOS-convex moment optimization and applications via exact semi-definite programs

This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.

Model Predictive Bang-Bang Controller Synthesis via Approximate Value Functions

In this paper, we propose a novel method for addressing Optimal Control Problems (OCPs) with input-affine dynamics and cost functions. This approach adopts a Model Predictive Control (MPC) strategy, wherein a controller is synthesized to handle an approximated OCP within a finite time horizon. Upon reaching this horizon, the controller is re-calibrated to tackle another approximation of the OCP, with the approximation updated based on the final state and time information. To tackle each OCP instance, all non-polynomial terms are Taylor-expanded about the current time and state and the resulting Hamilton-Jacobi-Bellman (HJB) PDE is solved via Sum-of-Squares (SOS) programming, providing us with an approximate polynomial value function that can be used to synthesize a bang-bang controller.

Recurrent Dissipativity-Based Inequalities for Controller Design

Using a new set of semidefinite constraints called recurrent dissipativity-based inequalities (DBIs), this letter presents an iterative procedure to design polynomial feedback control laws for polynomial nonlinear systems, let it be a static state feedback or a linear static output feedback (SOF) controller one needs to determine. In addition to that, the problem of linear SOF design for linear time-invariant (LTI) systems is solved as well. In the case of polynomial systems, we use sum-of-squares (SOS) programming for controller design and to provide an estimate of the closed-loop domain of attraction. In the case of LTI models, a set of linear matrix inequalities (LMIs) is employed in an iterative strategy to determine a stabilizing gain. Numerical simulations on a few examples borrowed from literature are provided in order to emphasize the advantages of our new dissipativity-based control (DBC) method.

Stability analysis in rice landraces grown under dry Direct seeded condition

In the present study the genotype x environment interaction and stability performances of 15 rice landraces in Rabi season during 2017-18 to 2021-22 at Agricultural Research Station, Tamil Nadu Agricultural University, Tamil Nadu, India were evaluated. Eberhart - Russel model was used for stability analysis in rice. In this model, genotype mean across environments, regression of genotype on environmental index and function of the squared deviation from the regression were determined. Mean sum of squares against environment source of variation as well as linear component of environment were significant for all the studied characters suggesting a presence of variation among the environments tested. The linear component of genotype x environment interaction was also found significant for all the traits studied which indicated presence of significant differences between the genotypes. The landraces Kallurundaikar and Kattanur responded favourably to better environments but gave poor yield in an unfavourable environment. Hence, they are suitable for favourable environments. Sivappuchithiraikar and Mysore malli recorded the highest grain yield with bi value closer to unity and non-significant deviation from regression. They were reported as stable genotypes and recommended for cultivation in southern districts of Tamil Nadu. Bangladesh J. Bot. 52(4): 941-948, 2023 (December)

Impact of Adoption of Advanced Agricultural Technologies on Farmer’s Income: Evidence from Central Punjab, Pakistan

Technology has significantly enhanced farmers' lives, primarily by facilitating access to market information, thereby mitigating the exploitation of farmers. Over the past century, farming methods and techniques have undergone numerous advancements and considerable changes. The main objective of the current research was to estimate the effect of farmer's socioeconomic characteristics and advanced technological adoption on farmer's income. For this purpose, Sargodha District was selected, and data collection involved conducting a survey. A well-structured questionnaire was distributed among 200 farmers across 6 Tehsils in the district of Sargodha. The study utilized an econometric model of Ordinary Least Squares (OLS), opting for a linear function to minimize the sum of squares of differences among observed values. According to the results of the study, there is a positive relationship between the adoption of advanced technology and agricultural income. It underscores the pivotal role that governments and agricultural welfare organizations should play a key role in technological advancement in agriculture. Increasing education can enhance the technology adoption process, while a necessary sensitization campaign is essential to facilitate the transition from traditional techniques to advanced techniques.

Genotype x environment interaction in hybrids and parents of Sesame (Sesamum indicum L.)

Pooled analysis of variance for phenotypic stability of 56 F1 hybrids along with 8 parents using 8 x 8 diallel mating design for seed yield, its components and oil content in sesame were assessed in four different environments in the Marathwada region of Maharashtra. Genotype x Environment (G x E) interaction was significant for all the characters except for days to maturity. The environment (linear) component of variance was significant for all the characters except oil content against pooled error, indicating the presence of macro-environmental differences under all four environments. The mean sum of squares due to G x E (linear) was significant for days to 50% flowering, days to maturity, plant height, number of capsules per plant, length of capsule, number of seeds per capsule, 1000 seed weight and seed yield per plant indicating variation in the performance of genotypes due to the regression of genotypes on the environment. The mean sums of squares due to pooled deviation (non-linear) were significant for most characters, suggesting the role of unpredictable causes affecting stability. However, the magnitude of non-linear components was lower for most of the characters than G x E (linear) variance. Among the crosses, TBS-10 x V-29, TBS-7 x TBS-12 and R-20 x R-09 were the most stable for seed yield per plant over a wide range of environments. These stable crosses also showed stable performance for most of the yield attributing traits and oil content.

Mass Transfer of Natural Dye Extraction and the Degradation Rate

This research studied the effect of temperature on the mass transfer rate and degradation rate of natural dye extraction. As the representative, two natural dyes extracted from Senduduk and Susupan Gunung plants were employed. Senduduk and Susupan Gunung are weed plants that are often found in peatlands. Susupan Gunung is easily found in South Kalimantan, and local people use it as a natural dye for handicraft products. Senduduk is often found in the South Sumatra area also used as a natural dye for traditional fabrics. Senduduk and Susupan Gunung were extracted in a three-neck flask using water as a solvent at temperatures of 50°, 70°, and 90°C. Samples of the extract solution were taken at certain times until the tannin concentration was constant. The tannin concentration in the solution was analyzed using volumetric methods. Experimental data consists of tannin concentrations in solution at various times. The mass transfer coefficient, equilibrium constant, and tannin degradation rate constant were determined by minimizing the sum square of errors between experimental data and the model. Senduduk contains more tannin (0.0094 g/cm3 solution) than Susupan Gunung (0.0079 g/cm3 solution), and Senduduk has a higher mass transfer coefficient than Susupan Gunung. The higher the temperature, the greater the rate of tannin degradation. During the extraction process, extract of Senduduk and Susupan Gunung leaves are stable at 50°C, which tends to degrade faster at higher temperatures.

Estimation of Nature and Magnitude of Genetic Variability, Heritability and Genetic Advance for Yield and Its Contributing Traits in Wheat (Triticum aestivum L.)

The present research investigation was conducted at AKS University, Satna, Madhya Pradesh during rabi Season of 2022. Altogether 20 wheat genotypes were grown under RBD design with three replications. In the present study various parameters of genetic variability, broad sense heritability and genetic advance estimates of wheat crop. Analysis of variance revealed that the mean sum of square showed significant due to genotypes for all the 12 characters studied. Analysis of variance for the design of experiment revealed highly significant difference for all the characters which expressed huge amount of diversity among 20 genotypes of wheat. Phenotypic coefficients of variation (PCV) were higher than genotypic coefficients of variation (GCV) for all the characters. The maximum GCV and PCV were observed for ear length (cm) (26.37 & 27.04) followed by number of productive tillers per plant (23.72 & 25.65) and test weight (21.76 & 22.57 g). The maximum value of heritability was shown by number of spikelets per ear (98.05), number of seed per spike (97.59), ear length (95.12cm), test weight (92.99 g), biological yield per plant (91.99 g), plant height (89.86 cm), seed yield per plant (87.28 g), days to maturity (86.67) and number of productive tillers per plant (85.56). The highest genetic advance was found for number of productive tillers per plant (45.20), number of seed per spike (43.60), test weight (43.24 g), seed yield per plant (37.37 g), no of ears par plant (34.99) and number of spikelets per ear (31.49). High heritability coupled with high genetic advance for the characters number of productive tillers per plant, no of ears per plant, ear length, number of spikelets per ear, number of seed per spike, test weight and seed yield per plant which indicated that above characters were governed by additive gene action and direct selection may be useful for further improvement program.

Genetic Characterization and Character Association Study in Bread Wheat (Triticum aestivum L.) for Grain Yield and Yield Attributing Traits

Present research work was carried out at the research farm of Department of Genetics and Plant Breeding, Rama University, Kanpur to investigate the pattern of genetic variability and heritability in the elite genotypes of wheat germplasm. In current study 27 wheat genotypes including checks entries viz., HD 2967 AND PBW 152 have been evaluated in randomized block design in 3 replication and the observations have been recorded on the eleven metric traits viz., Days to 50% flowering, Days to maturity, Flag leaf area (cm2), Tillers per plant, plant height (cm), spike length (cm), Spikelet per spike, Grain yield per plant (g), Biological yield per plant (g), Test weight (g) and Harvest index (%). The analysis of variance showed significant mean sum of squares due to all traits under study. High estimates of heritability were observed for all of the traits viz., grain yield per plant (98.90%) followed by test weight (95.53%) and harvest index (94.88 %), whereas, high estimates of genetic advance (>20%) in per cent over mean was recorded for the traits grain yield per plant (44.26%) followed by spike length (36.12%) and Test weight (28.51%), High heritability accompanied with high genetic advance for the traits showed additive gene action and the selection for such traits would be rewarding.

Fault Classification and Diagnosis Approach Using FFT-CNN for FPGA-Based CORDIC Processor

Within the realm of digital signal processing and communication systems, FPGA-based CORDIC (Coordinate Rotation Digital Computer) processors play pivotal roles, applied in trigonometric calculations and vector operations. However, soft errors have become one of the major threats in high-reliability FPGA-based applications, potentially degrading performance and causing system failures. This paper proposes a fault classification and diagnosis method for FPGA-based CORDIC processors, leveraging Fast Fourier Transform (FFT) and Convolutional Neural Networks (CNNs). The approach involves constructing fault classification datasets, optimizing features extraction through FFT to shorten the time of diagnosis and improve the diagnostic accuracy, and employing CNNs for training and testing of faults diagnosis. Different CNN architectures are tested to explore and construct the optimal fault classifier. Experimental results encompassing simulation and implementation demonstrate the improved accuracy and efficiency in fault classification and diagnosis. The proposed method provides fault prediction with an accuracy of more than 98.6% and holds the potential to enhance the reliability and performance of FPGA-based CORDIC circuit systems, surpassing traditional fault diagnosis methods such as Sum of Square (SoS).

Developing customized fuel models for shrub and bracken communities in Galicia (NW Spain)

Geospatial fire behaviour and fire hazard simulators, fire effects models and smoke emission software commonly use standard fuel models in order to simplify data collection and the inclusion of complex fuel scenarios. These fuel models are often mapped using remotely sensed data. However, given the great complexity of fuelbeds, with properties that vary widely in both time and space, the use of these standard fuel models can greatly limit accurate fuel mapping. This affects fuel hazard assessment, fuel reduction treatment plans, fire management decision-making and evaluation of the environmental impact of wildfire. In this study, we developed unique customized fire behaviour fuel models for shrub and bracken communities, by using k-medoids clustering analysis based on both fuel structural characteristics and potential fire behaviour. We used an original database of 722 destructive sample plots in nine different shrub and bracken communities covering the entire distribution area in Galicia (NW Spain), one of the regions in Europe most affected by forest fires. Measurements of cover, height and fuel fractions loads differentiated by size and vegetative state (live or dead) were used to estimate the potential rate of fire spread with five different models including fireline intensity, heat per unit area and the flame length for each sampling site and considering extreme environmental conditions. The optimal number of clusters was established by combining practical knowledge about the shrubland communities under study and their associated fire behaviour, with maximization of the mean value of the silhouette variable and minimization of the within-cluster sum of squares. The structural characteristics of the medoids derived from the analysis were associated with each of the proposed customized fuel models. Finally, a simple dichotomous classification based only on shrub height was developed to enable construction of spatially explicit fuel model maps based on remotely sensed data. Thus, the methodology applied allows generation of a more realistic representation of fuel distribution in the landscape, based on fuel structure measurements of natural regional ecosystems rather than on the use of standard models. We believe that the proposed methodology is generally applicable to communities composed of other shrub and fern species in different biogeographical regions.

On linear stabilization of a class of nonlinear systems in a critical case

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

Estymacja parametrów asymetrycznych procesów niegaussowskich o średniej kroczącej metodą maksymalizacji wielomianu stochastycznego PMM

In this paper consider is the application of the Polynomial Maximization Method PMM to find estimates of the parameters of non-Gaussian Moving Average model. This approach is adaptive and is based on the analysis of higher-order statistics. The case of asymmetry of distributions of Moving Average of the stochastic processes is also considered. It is shown that the asymptotic variance of estimates of the Polynomial Maximization Method (2nd order) have such analytical expressions, whose allow to finding estimates and analyzing their uncertainties. Above approach can be significantly less than the variance of the classic estimates based on minimizing the Conditional Sum of Squares or Maximum Likelihood (in the Gaussian case). The increase of accuracy depends on the values of the coefficient’s asymmetry and the kurtosis of residuals. The results of statistical modeling by the Monte Carlo Method confirm the effectiveness of the proposed approach.

Frequency of heterotic hybrids in relation to general combining ability of parents in sweet corn.

The success of developing prominent hybrids directly depends on the selection of parents with good combining ability which can transfer desirable genes with additive effects to their progeny. The data of 42 hybrids generated using 7×7 full diallel design; their seven parents along with three check hybrids were subjected to combining ability analysis from the experiment that was carried out during rainy season 2019. The analysis of variance showed significant general combining ability, and specific combining ability mean sum of squares for all the thirteen characters studied. It is obvious from the results that three lines (SC Sel 2, SC Sel 1 and SC Sel 3) showed high overall general combining ability status, suggesting these lines as good general combiners across traits. Eighteen hybrids had high overall specific combining ability status, while nearly 52% (22 hybrids), 57% (24 hybrids) and 47% (20 hybrids) of crosses showed high overall mid-parent, better-parent and standard heterosis. The unique superiority of crosses involving high overall general combiner parent in the crosses highlighted the importance of using such parents to realize high heterotic crosses. A non-linear relationship between high overall specific combining ability status and heterotic status of hybrids was noticed. The probability of obtaining a cross with high standard heterosis was more with employing parents with high general combining ability status.

Asymmetric Effects of Exchange Rate Shocks on Urban Household Health Expenditures in Iran

Background: The evidence of Iran's economy shows that many economic variables, including household health expenditures, can be affected by exchange rate shocks. According to previous studies, the increase in the exchange rate from two channels of the general increase in prices, as well as the increase in the price of medicines and imported medical supplies, increases health expenditures. This study investigated the asymmetric effects of the unofficial exchange rate on the health expenditures of urban households in Iran.
 Methods: The present applied study investigated the short-run and long-run effects of positive and negative shocks of the unofficial exchange rate, gross domestic product (GDP), and CO2 emissions on the health expenditures of urban households during 1971-2019 using time series data. The data were collected from the database of the Statistics Center and the Central Bank. To investigate the reliability of data, generalized Phillips-Prone and Dickey-Fuller tests and to check the stability of the coefficients, cumulative sum and cumulative sum squares were used. Also, to check the short-run dynamics and adjustment towards the long-run, the error correction model was estimated. The regression model was estimated using the Nonlinear Autoregressive Distributed Lag approach by Eviews (12.0) software. After estimation, diagnostic tests including Wald test, normality, heterogeneity of variance and autocorrelation were performed.
 Results: The results showed that the increase in exchange rate has a positive and significant effect on household health expenditures, its reduction has no significant effect Therefore, the effect of exchange rate shocks was asymmetric. Accordingly, 1% increase in the unofficial exchange rate led to a 0.237 % increase in household health expenditures in the short-run and 0.55 % increase in the long-run. In addition, 1% increase in GDP led to 0.49 % increase in household health expenditures in the short-run and 1.57% increase in the long-run. Finally, 1 % increase in CO2 emissions led to a 1.49 % increase in household health expenditures in the long-run.
 Conclusion: Based on the results, after the increasing exchange rate shocks and possible future declines, due to the persistence of the shocks effects for up to three years, it is suggested not to stop supportive policies for providing health expenditures, especially for low-income groups.

The Existence and Stability of Limit Cycle of Some Special Types of 2D Autonomous Dynamic Systems

This article is focusing on the problem about limit cycles of 2D autonomous dynamics, which is a part of the famous Hilbert’s 16th problem. There are mainly three results on this paper. The first two results discuss the cases with sum of squares of the variables. Through polar substitution and the mathematics analysis on the explicit limit cycle obtained by Cramer’s Rule, the first result reveals the uniqueness and stability of the limit cycle of a specific form for 2D autonomous dynamic system. For the second part of the results, by changing the coefficients of the system of the first result, it is found that the stability of limit cycle remains unchanged on the phase portrait. The third part gives a special polynomial with terms of general odd degrees for Liénard equation, which can be rewritten as a case of 2D autonomous dynamic system. Without explicit expression of the limit cycle, the Poincaré–Bendixson method and the Liénard theorem help to give the conclusion on both uniqueness and stability of this system.

Investigation of the influence of various factors on the wear of rod depth pumps

In the article, the influence of various factors on the wear of rod depth pumps is investigated. The method of variance analysis of mathematical statistics is applied here. The main factors selected are: the diameter of the pump, the percentage of sand, the amount of water, the amount of oil, the lifting height of the liquid. The task of the study is to find the degree of influence of the selected factors, while it is assumed that the pump supply coefficient decreases only to 0.6. In this case, the main values are in the following numbers of levels: pump diameter, mm; 32; 43; 44; 56; 70; percentage of sand, %; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1; the amount of water, m3/day; 10; 20; 30; 40; 50; the amount of oil, m3/day 0.5; 1; 1.5; 2; 2.5; 3; 3.5; 4; 4.5; 5; liquid lifting height, m; 200, 400, 600, 800, 1000 m. In the article, the variable X=Snt, its average value and the concepts of variances are explained. Various numbers of degrees of freedom are presented. Here the smallest value is 8 and the largest value is 81. Table 2 presents the variability abilities, the number of degrees of freedom, the sum of squares and the variance estimation. Here, the largest variance value is 0.560 and the smallest was 0.065. The exact value of the residual mean squares was 0.084. In the case of, the amou interactions of all the analyzed factors, the amount of water in the product, its lifting heightnt of water and the diameter of the pump matter.