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In case of multicollinearity, PRESS statistics has been proposed to be used in the selection of the ridge biasing parameter of the ridge estimator which is introduced as an alternative to BLUE. This newly proposed PRESS statistic for the ridge estimator, CPRESS k , depends on the conditional ridge residual and can be computed once at a time by fitting the linear mixed model with all the observations. We also define R RidPred 2 statistic to evaluate the predictive ability of the ridge fit. Since the PRESS statistic for the BLUE is a special CPRESS k statistic, we indirectly also give closed form solution of the PRESS statistic for the BLUE. Then, we compared the predictive performance of the linear mixed model via the statistics CPRESS k , G C V k and C p by considering a real data analysis and a simulation study where the optimal ridge biaisng parameter is obtained by minimizing each statistic. The study shows that the ridge predictors improve the predictive performance of a linear mixed model over BLUE in the presence of multicollinearity and each statistic gives a different optimum ridge biasing value and they show the best predictive performance at their optimum ridge biasing values. In addition, the simulation study has shown that the intensity of variance and multicollinearity is effective in determining the optimum ridge biasing value and this optimum ridge biasing value is effective on the superiority of the predictive performance of ridge estimator over BLUE. [ABSTRACT FROM AUTHOR]

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures. [ABSTRACT FROM AUTHOR]

It is a fact that the performance of most crop genotypes varies across environments due to genotype‐environment interactions. Sharing knowledge about maize genotypes that are well‐adapted and high‐yielding in specific environments is essential for smallholder farmers to boost their productivity. A field trial was established in three locations during the 2019 and 2020 growing seasons to identify and select better‐adapted high‐yielding maize genotypes for production. A total of 10 improved varieties were evaluated using a randomized complete block design with three replications. The AMMI analysis of variance showed grain yield significantly (p ≤ 0.001) influenced by genotype‐environment interaction. This suggests that different genotypes showed varying grain yield performance across environments, implying the opportunity to choose effective genotypes suitable for both specific and broad adaptability. Moreover, the combined analysis of variance showed that genotype‐environment interaction significantly influenced plant height, ear height, dry biomass yield, and hundred‐kernel weight. The sum of squares factor explained that grain yield variation was contributed by the genotype (71.22%), followed by the environment (16.54%) and the genotype‐environment interaction effect (12.24%) of the total variation. Genotypes BH‐661, BH‐549, and BH‐546 demonstrated superior performance and stability, surpassing the national maize mean productivity and indicating broad adaptability. The AMMI analysis, GSI, and GGE biplot confirmed their robustness and high yield potential, especially for BH‐661 and BH‐549, making them optimal for variable conditions. Therefore, BH‐661 and BH‐549 are selected as stable genotypes for production in all agroecologies suitable for maize in the Dawuro zone. [ABSTRACT FROM AUTHOR]

Sea surface temperature (SST) is a key variable in the study of the global climate system and one of the important parameters in the process of air–sea interaction. Therefore, the demand for SST is developing towards high quality and high precision. In this paper, the ability of the multi-layer perceptron (MLP) optimized by the Aquila optimizer (AO) to solve nonlinear problems and the advantages of the deep belief network (DBN) to effectively process complex data are used to construct the IMLP-DBN inversion algorithm. The algorithm takes into account the influences of atmospheric conditions and satellite zenith angles. The data set is the infrared remote sensing data of the moderate resolution imaging spectroradiometer (MODIS) and the actual measurement data of buoys on sunny days and few clouds. Analysis of the inversion results shows that the root mean square error (RMSE) of the inversion value and the measured value is 0.14, and the sum of square errors (SSE) is 0.78. Compared with the MOD28 product data, the RMSE and SSE of the inversion are reduced by 66% and 24%, respectively. [ABSTRACT FROM AUTHOR]

This paper presents an innovative method to tackle the predictive-control challenges associated with large-scale fuzzy polynomial systems comprising interconnected polynomial fuzzy systems. This study models large-scale nonlinear fuzzy systems in a polynomial framework, which can reduce the number of fuzzy rules. We derive the conditions for controller synthesis in the main theorem using the Lyapunov theory and sum-of-squares technique. Simulation results confirm the validity and efficiency of this approach. [ABSTRACT FROM AUTHOR]

Tensor spectral p -norms are generalizations of matrix induced norms. Matrix induced norms are an important type of matrix norms, and tensor spectral p -norms are also important in applications. We discuss some basic properties of tensor spectral p -norms. We extend the submultiplicativity of the matrix spectral 2-norm to the tensor case, based on which we give a bound of the tensor spectral 2-norm and provide a fast method for computing spectral 2-norms of sum-of-squares tensors. To compute tensor spectral p -norms, we propose a higher order power method. Experiments show the high efficiency of the proposed methods and numerical results on spectral p -norms of random tensors are also given. [ABSTRACT FROM AUTHOR]

Barley (Hordeum vulgare L.) is a major grain crop farmed in Ethiopia throughout the long rainy season (Meher) and the short rainy season (Belg) of the year. Barley genotypes were subjected to multi‐environment experiments in six different settings to identify stable genotypes and estimate the impact of genotype × environment interaction (GEI) on grain production. In each area, the field experiment was conducted from mid‐July to January during the primary cropping season of 2021. Three replications of a randomized complete block design were used to set up the trials. According to the additive main effects and multiplicative interaction (AMMI) study, genotype (18.19%), GEI (22.98%), and environment (58.83%) all had an impact on the major treatment sum of squares. The more variance attributed to the environments is a sign of environmental diversity. Given that the two interaction principal component analysis (IPCAs) accounted for 76.94% of the interaction sum of squares, they were sufficient for cross‐validation of the grain yield variance explained by GEI. In contrast to the GGE biplot approaches, which indicated genotypes G12, G3, and G9 as stable and high‐yielding genotypes throughout the environments, the AMMI stability value identified genotypes G3, G12, and G9 as high yielding with stable performance across environments. In general, the GGE biplot and AMMI analysis models demonstrated that genotypes G12, G3, and G9 were stable and yielded well, making G3 acceptable for cultivation in a wider range of environments and G12 and G9 suitable for release. Core Ideas: The environment explained the greatest portion of the variation in grain yield in this study (58.83%), followed by the GEI (22.98%) and genotype (18.19%).Environment Bule is more effective at discriminating genotypes and is representative of the test environments.additive main effects and multiplicative interaction (AMMI) and GGE biplot models identified high yielding and stable genotypes.GGE biplot and AMMI models demonstrated that genotypes G12, G3, and G9 were stable and high yielding.G3 acceptable for cultivation in a wider range of environments and G12 and G9 suitable for release. [ABSTRACT FROM AUTHOR]

Poor irrigation water quality can mar agricultural productivity. Traditional assessment of irrigation water quality usually requires the computation of various conventional quality parameters, which is often time-consuming and associated with errors during sub-index computation. To overcome this limitation, it becomes critical, therefore, to have a visual assessment of the irrigation water quality and identify the most influential water quality parameters for accurate prediction, management, and sustainability of irrigation water quality. Therefore, in this study, the overlay weighted sum technique was used to generate the irrigation water quality (IWQ) map of the area. The map revealed that 29.2% of the area is suitable for irrigation (low restriction), 41.7% is moderately suitable (moderate restriction); and 29.1% is unsuitable (high restriction), with the irrigation water quality declining towards the central-southeastern direction. Multilayer perceptron artificial neural networks (MLP-ANNs) and multiple linear regression models (MLR) were integrated and validated to predict the IWQ parameters using Cl−, HCO3− SO42−, NO3−, Ca2+, Mg2+, Na+, K+, pH, EC, TH, and TDS as input variables, and MAR, SAR, PI, KR, SSP, and PS as output variables. The two models showed high-performance accuracy based on the results of the coefficient of determination (R2 = 0.513–0.983). Low modeling errors were observed from the results of the sum of square errors (SOSE), relative errors (RE), adjusted R-square (R2adj), and residual plots, further confirming the efficacy of the two models; although the MLP-ANNs showed higher prediction accuracy for R2. Based on the sensitivity analysis of the MLP-ANN model, HCO3, pH, SO4, EC, and Cl were identified to have the greatest influence on the irrigation water quality of the area. This study has shown that the integration of GIS and machine learning can serve as rapid decision-making tools for proper planning and enhanced agricultural productivity. [ABSTRACT FROM AUTHOR]

We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are called generalized strips). This class includes the strip [ a , b ] × R which was studied by Marshall in (Proc Am Math Soc 138(5):1559–1567, 2010). A denominator-free Nichtnegativstellensätz holds true on a generalized strip when the width of the generalized strip is constant and fails otherwise. As a consequence, we confirm that the standard hierarchy of semidefinite programming relaxations defined for the compact case can indeed be adapted to the generalized strip with constant width. For polynomial optimization problems on the generalized strip with non-constant width, we follow Ha-Pham's work: Solving polynomial optimization problems via the truncated tangency variety and sums of squares. [ABSTRACT FROM AUTHOR]

In 1972, John Brillhart described an algorithm for expressing a prime p ≡ 1 (mod 4) as the sum of two squares. Brillhart's algorithm, which is based on the Euclidean algorithm, is simplicity itself. However, Brillhart's proof of his algorithm's correctness uses several previous results, and subsequent simplifications of his argument still retain something of an air of mystery. We provide a geometric interpretation of Brillhart's algorithm, which not only proves that it works, but also sheds light on a surprising palindromic property proved by Perron, and used by Brillhart in his proof. [ABSTRACT FROM AUTHOR]

The performance differences in cassava genotypes arising from genotype vs. environment interactions (G × E) often lead to responses that are significantly lower than expected for selection. The objective of this study was to evaluate different stability methods, both parametric and non-parametric, such as additive main-effects and multiplicative interaction (AMMI), main effect of genotypes plus G × E (GGE), and weighted average of absolute scores (WAASB), in order to quantify the G × E in multi-environmental trials. A total of 12 genotypes were assessed across 12 environments using a completely randomized block design, with three replicates for traits such as fresh root yield (FRY) and dry matter content in the roots (DMC). The data were subjected to analysis of variance and the Scott Knott test (p < 0.05). The sum of squares (SQ) of genotypes, environment, and G × E effects were equally distributed for FRY, whereas for DMC, these effects accounted for 64.1%, 21.9%, and 13.8% of the SQ, respectively, indicating a lower environmental effect on this characteristic. Using the AMMI, GGE, and WAASB methods, genotypes with high agronomic performance and stability for FRY (BR11-34–41 and BR11-34–69) (> 32 t ha−1) and DMC (BRS Novo Horizonte, BR12-107–002, and BR11-24–156) (> 37%) were identified. The broad-sense heritability ( h 2 ) for FRY and DMC was estimated to be 0.45 and 0.75, respectively. Approximately 72% of the methods identified BRS Novo Horizonte as the genotype with the highest stability and performance for DMC, while 47% identified genotypes BR11-34–41 and BR11-34–69 for FRY and intermediate DMC. Genotype BR11-24–156 exhibited high static stability according to 50% of the methods. Significant correlations were observed between stability and agronomic performance across the different methods, enabling the formation of groups based on stability concepts. Additionally, it was found that two mega-environments existed for FRY, whereas DMC displayed a single mega-environment with similar patterns, indicating an absence of G × E. We identified superior genotypes that could be promoted to national performance trials to develop stable cultivars with better yield attributes in cassava. [ABSTRACT FROM AUTHOR]

Resilience describes the capacity of systems to react to, withstand, adjust to, and recover from disruptive events. Despite numerous metrics proposed to quantify resilience, few studies predict these metrics or the restoration time to nominal performance levels, and these studies often focus on a single domain. This paper introduces three methods to model system performance and resilience metrics, which are applicable to various engineering and social science domains. These models utilize reliability engineering techniques, including bathtub-shaped functions, mixture distributions, and regression analysis incorporating event intensity covariates. Historical U.S. job loss data during recessions are used to evaluate these approaches' predictive accuracy. This study computes goodness-of-fit measures, confidence intervals, and resilience metrics. The results show that bathtub-shaped functions and mixture distributions accurately predict curves possessing V, U, L, and J shapes but struggle with W and K shapes involving multiple disruptions or sudden performance drops. In contrast, covariate-based models effectively track all curve types, including complex W and K shapes, like the successive shocks in the 1980 U.S. recession and the sharp decline in the 2020 U.S. recession. These models achieve a high predictive accuracy for future performance and resilience metrics, evidenced by the low sum of square errors and high adjusted coefficients of determination. [ABSTRACT FROM AUTHOR]

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $ y=q(x) $ y = q (x) , $ q(x)\in \mathbb R[x] $ q (x) ∈ R [ x ] , $ \deg q\geq ~3 $ deg ⁡ q ≥ 3 and $ yx^\ell =1 $ y x ℓ = 1 , $ \ell \in \mathbb N\setminus \{1\} $ ℓ ∈ N ∖ { 1 }. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: .; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. .; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: .], and improve Fialkow's bound to $ \deg q-1 $ deg ⁡ q − 1 (resp. $ \ell +1 $ ℓ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $ k\deg q $ kdeg ⁡ q (resp. $ k(\ell +1) $ k (ℓ + 1)) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1) for even degree sequences, while for odd ones they are $ k\deg q-\big \lceil \frac {\deg q}{2} \big \rceil $ kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. $ k(\ell +1)-\big \lfloor \frac {\ell }{2} \big \rfloor +1 $ k (ℓ + 1) − ⌊ ℓ 2 ⌋ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $ y=x^\ell $ y = x ℓ , $ \ell =2,3 $ ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve $ y=x^2 $ y = x 2 . [ABSTRACT FROM AUTHOR]

The proton exchange membrane fuel cell (PEMFC) converts chemical energy into electricity without pollution or noise. PEMFC has a stable electrolyte, reliable, fast start‐up, and lightweight portability. Optimal PEMFC parameter prediction is difficult. Due to inaccurate estimation, many meta‐heuristic models are inaccurate. Particle swarm, Ant colony, and grey wolf algorithms are tested for fuel cell parameter estimation. These traditional algorithms benefit from chaotic maps for exploitation and exploration. Analyzing V–I and P–I polarization curves, hybrid approaches estimate unknown parameters. Results are compared with others to support the identifying strategy. This project examined SR‐12 and NedSstack PS‐6 PEMFC stacks. The objective function was shown by the sum of square errors between the experiments, and once an appropriate map is identified, the single chaotic map inclusion method can perform better than the multiple chaotic map scheme, and the "Iterative" approach in chaotic swarm is the most promising for multidimensional challenges. [ABSTRACT FROM AUTHOR]

This paper presents a novel maximum power point tracking control for a stand-alone photovoltaic (PV) system based on a robust polynomial static output feedback control law subject to input saturation. In detail, a DC/DC boost converter is used to regulate the load and extract the maximum power from the photovoltaic panel. First, a polynomial fuzzy model is used to represent the photovoltaic system. Then, as this control method is based on a reference model, a regression plane is used to generate the desired trajectory representing the optimal dynamics where the PV system supplies maximum power. Then, in order to reduce the number of required sensors, a polynomial output feedback controller was developed, in which the problem of converter performance degradation resulting from duty cycle saturation was avoided by using a saturated control approach. The controller gains have been obtained by solving a sum-of-squares optimization problem, where the H ∞ performance criterion is applied to guarantee the stability of the closed-loop system while achieving an optimal rejection level of external disturbances. To evaluate the performance of the suggested controller, a number of simulations and comparisons were carried out in MATLAB/Simulink environment and under various scenarios of weather conditions. [ABSTRACT FROM AUTHOR]

In this paper, an observer-based tracking control problem is considered for a polynomial system with parameter uncertainties and external disturbances. Noting that some states are difficult to measure in practice, a polynomial state observer is employed to obtain the unknown states. Then a robust observer-based tracking controller is designed to force the states of the polynomial uncertain system to follow the ones of the reference model and satisfy $ H_\infty $ H ∞ performance. By using a full-state-dependent homogeneous polynomial function, the solvability condition is derived for the observer-based tracking control strategy, which reduces conservatism. Sum of squares technique is utilised to calculate the corresponding observer and controller, effectively dealing with the calculation problem in polynomial systems. Simulation examples are given to illustrate the effectiveness of the presented control strategy. [ABSTRACT FROM AUTHOR]

Solar pond is one of the latest technologies that have been used to produce fresh water in different countries. The studied solar pond consists of a metal floor and glass walls. In this research, a mathematical model is presented with the aim of estimating salt deposition. The results show that proposed model has high accuracy. Results show that sum of square error and relative error of proposed equation are equal to 3.048E−14 and 0.11214, respectively. In addition, the p-value and square error of proposed model have been validated. Results show that p-value of temperature and shear stress has a significant contribution in salt deposition. This research shows that R-Squared of proposed model is equal to 0.9359. This value indicates good validity of the proposed model. [ABSTRACT FROM AUTHOR]

This work represents a natural coalescence of two important lines of work — learning mixtures of Gaussians and algorithmic robust statistics. In particular, we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving a type of dimension-independent polynomial identifiability — which we call robust identifiability — through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation. [ABSTRACT FROM AUTHOR]

AbstractThis study investigates the equilibrium and kinetics of copper and cadmium adsorption from aqueous solutions using groundnut husk. Characterization of the groundnut husk reveals various functional groups and a porous morphology. Isothermal studies were conducted, and different models were evaluated to fit the data. The Langmuir model exhibits the best fit for copper adsorption, while the Temkin and Dubinin–Radushkevich (DR) isotherm models show superior fitting for cadmium adsorption. Results suggest that physisorption significantly influences both cases, with mean sorption adsorption energy lower than 8000 J/mole. Error analysis reveals that the Freundlich and Temkin models demonstrate maximum adsorption capacity, with lower Chi-square (ꭓ2), Sum of Square Errors (SSE), and SAE compared to Langmuir and DR models. Additionally, intra-particle diffusion was analyzed using the Weber and Morris equation. Evaluation of model parameters indicates that cadmium demonstrates better nonlinear fitting of data compared to copper. FTIR and SEM analysis confirm the involvement of various functional groups in adsorption and reveal changes in surface distribution post-adsorption due to copper and cadmium presence. [ABSTRACT FROM AUTHOR]

Substitution boxes, or S-boxes, are one of the most important mathematical primitives in modern symmetric cryptographic algorithms. Given their importance, in the past decades, they have been thoroughly analyzed and evaluated by the academic world. Thus, a lot of desirable characteristics a given S-box should possess have been found. This includes, as much as possible, higher nonlinearity and algebraic degrees as well as, as much as possible, lower values of differential uniformity, autocorrelation and sum of squares indicator values. In this work, we use power mappings over G F (2 8) to generate, enumerate and evaluate all bijective S-boxes yielded by pentanomials of the form f (x) = x a + x b + x c + x d + x e given 0 < a < b < c < d < e < 256 . We find a total of 152,320 different bijective S-boxes, which are further classified into 41,458 different groups in terms of the aforementioned characteristics as well as the number of their fixed points. Having this data, an S-box designer can easily generate a bijective substitution S-box with parameters of their choice. By using pentanomials, we show how we can easily construct S-boxes with cryptographic properties similar to those found in some popular S-boxes like the Kuznyechik S-box proposed by the Russian Federation's standardization agency as well as the Skipjack S-box proposed by the National Security Agency of the USA. [ABSTRACT FROM AUTHOR]

In this work we complement the description of the extreme rays of the 6 × 6 copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of subsets of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those subsets which are essential, i.e., which are not embedded in the boundary of other subsets. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the 6 × 6 copositive cone. As an application, we construct an example of a copositive 6 × 6 matrix with all ones on the diagonal which does not belong to the Parrilo inner sum of squares relaxation K 6 (1). [ABSTRACT FROM AUTHOR]

For integer k ≥ 2 , a graph G is called k -leaf-connected if | V (G) | ≥ k + 1 and given any subset S ⊆ V (G) with | S | = k , G always has a spanning tree T such that S is precisely the set of leaves of T. Thus a graph is 2-leaf-connected if and only if it is Hamiltonian-connected. The first Zagreb index M 1 (G) is equal to the sum of squares of the degrees of the vertices of the underlying molecular graph G. The reciprocal degree distance (RDD), defined as vertex-degree-weighted sum of the reciprocal distances, that is, RDD (G) = ∑ v i ≠ v j deg G (v i) + deg G (v j) d G (v i , v j) , where deg G (v i) is the degree of the vertex v i in G and d G (v i , v j) denotes the distance between two vertices v i and v j in G. The forgotten topological index of G is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions in terms of the first Zagreb index, the reciprocal degree distance or the forgotten topological index for a graph to be k -leaf-connected, which generalizes two results of An (2022). All of these results are best possible. • A sharp bound on the first Zagreb index that ensures k-leaf-connected is given. • A sharp bound on reciprocal degree distance that ensures k-leaf-connected is given. • A sharp bound on the forgotten topological index that ensures k-leaf-connected is given. [ABSTRACT FROM AUTHOR]

Abstract.Spearman’s footrule is a widely used tool to discover the relationship between two continuous random variables, mainly because of its robustness, natural interpretation, and simplicity of calculation. In this article, we propose a new test statistic based on the sum of squares of pairwise Spearman’s footrule, to test the mutual independence of a high dimensional random vector. The large sample properties of the proposed test method are established. Simulation results show that the new test has higher power than existing test methods under some circumstances. An example with real life data is discussed to illustrate the usefulness and effectiveness of the proposed test statistic. [ABSTRACT FROM AUTHOR]

We aimed to evaluate the feasibility of non-contrast-enhanced T1 sequence in texture analysis of breast cancer lesions to predict their estrogen receptor status. The study included 85 pathologically proven breast cancer lesions in 53 patients. Immunohistochemical studies were performed to determine the estrogen receptor status (ER). Lesions were divided into two groups: ER + ve status and ER-ve status. Texture analysis using the second-order analysis features [The Co-occurrence matrix (11 features)] was applied on both T1 and dynamic contrast-enhanced (DCE) MRI images for each lesion. Texture features gained from both T1 and DCE images were analyzed to obtain cut-off values using ROC curves to sort lesions according to their estrogen receptor status. Angular second momentum and some of the entropy-based features showed statistically significant cut-off values in differentiation between the two groups [P-values for pre- and post-contrast images for AngSecMom (0.001, 0.008), sum entropy (0.003,0.005), and entropy (0.033,0.019), respectively]. On comparing the AUCs between pre- and post-contrast images, we found that differences were statistically insignificant. Sum of squares, sum variance, and sum average showed statistically significant cut-off points only on pre-contrast images [P-values for sum of squares (0.018), sum variance (0.024), and sum average (0.039)]. Texture analysis features showed promising results in predicting estrogen receptor status of breast cancer lesions on non-contrast T1 images. • Prediction of estrogen receptor status using MRI-based texture analysis. • Diagnostic accuracy of texture analysis using non-contrast and DCE images. • Best texture analysis features to predict the estrogen receptor status. [ABSTRACT FROM AUTHOR]

This paper introduces a novel formulation of the clustering problem, namely, the minimum sum-of-squares clustering of infinitely tall data (MSSC-ITD), and presents HPClust, an innovative set of hybrid parallel approaches for its effective solution. By utilizing modern high-performance computing techniques, HPClust enhances key clustering metrics: effectiveness, computational efficiency, and scalability. In contrast to vanilla data parallelism, which only accelerates processing time through the MapReduce framework, our approach unlocks superior performance by leveraging the multi-strategy competitive–cooperative parallelism and intricate properties of the objective function landscape. Unlike other available algorithms that struggle to scale, our algorithm is inherently parallel in nature, improving solution quality through increased scalability and parallelism and outperforming even advanced algorithms designed for small- and medium-sized datasets. Our evaluation of HPClust, featuring four parallel strategies, demonstrates its superiority over traditional and cutting-edge methods by offering better performance in the key metrics. These results also show that parallel processing not only enhances the clustering efficiency, but the accuracy as well. Additionally, we explore the balance between computational efficiency and clustering quality, providing insights into optimal parallel strategies based on dataset specifics and resource availability. This research advances our understanding of parallelism in clustering algorithms, demonstrating that a judicious hybridization of advanced parallel approaches yields optimal results for MSSC-ITD. Experiments on the synthetic data further confirm HPClust's exceptional scalability and robustness to noise. [ABSTRACT FROM AUTHOR]

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The study presents a hybrid model for recommending travel routes that takes into account multiple constraints. This model is based on a collaborative filtering recommendation algorithm and addresses the issue of disorganized travel route recommendations. To improve upon the k-mean clustering algorithm, the proposed model introduces the Dynamic Gaussian Kernel Density K-means algorithm. After the data was processed, the initial clustering center was determined and k-means clustering was performed. Subsequently, the travel route recommendation model was created by integrating various constraints. The study's proposed algorithm was compared with alternative algorithms, and the experimental results demonstrated superior performance across a range of datasets, with the minimum sum of squared errors and a running time of approximately 1.4 seconds - a noteworthy improvement. Comparative experiments were conducted on various forgetting coefficients in the model, and the forgetting coefficient with the lowest sum-of-squares of errors was selected to replace the existing one. Upon comparing the proposed research model with other models, it was found that the former had greater accuracy and recall, amounting to 98.1% and 96.8% respectively. This suggests that the proposed research model serves as a more efficient solution for travel route recommendation. [ABSTRACT FROM AUTHOR]

Hierarchical porous activated carbon derived from pleurotus eryngii was prepared by a one-step activation method. It was found that the specific surface area of the obtained sample increased with the increase in activation temperature (700–900 °C). The sample activated at 900 °C has a specific surface area of 2002.2 m2 g−1 and the highest specific capacitance (319 F g−1), which is mainly attributed to the high utilization rate of specific surface area brought by the hierarchical porous structure. The assembled PEK-900//PEK-900 capacitor measured a specific capacity of 258 F g−1 at a current density of 0.5 A g−1. After 10,000 cycles of charging and discharging, the specific capacitance increased by 10%. Based on the correlation analysis of experimental data between the specific capacitance and pore structural parameters, Lasso dimensionality reduction and binary linear regression were used to reveal the relationship between the two. The residual sum of squares obtained by this method decreased by 38.4% compared to the univariate linear regression, providing a simple and reliable theoretical method for predicting the capacitance performance of biomass carbon materials. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to predict stock market crashes identified by the CMAX approach (current index level relative to historical maximum) during periods of global and local events, namely the subprime crisis of 2008, the political and social instability of 2011 and the COVID-19 pandemic. Design/methodology/approach: Over the period 2004–2020, a log-periodic power law model (LPPL) has been employed which describes the price dynamics preceding the beginning dates of the crisis. In order to adjust the LPPL model, the Global Search algorithm was developed using the "fmincon" function. Findings: By minimizing the sum of square errors between the observed logarithmic indices and the LPPL predicted values, the authors find that the estimated parameters satisfy all the constraints imposed in the literature. Moreover, the adjustment line of the LPPL models to the logarithms of the indices closely corresponds to the observed trend of the logarithms of the indices, which was overall bullish before the crashes. The most predicted dates correspond to the start dates of the stock market crashes identified by the CMAX approach. Therefore, the forecasted stock market crashes are the results of the bursting of speculative bubbles and, consequently, of the price deviation from their fundamental values. Practical implications: The adoption of the LPPL model might be very beneficial for financial market participants in reducing their financial crash risk exposure and managing their equity portfolio risk. Originality/value: This study differs from previous research in several ways. First of all, to the best of the authors' knowledge, the authors' paper is among the first to show stock market crises detection and prediction, specifically in African countries, since they generate recessionary economic and social dynamics on a large extent and on multiple regional and global scales. Second, in this manuscript, the authors employ the LPPL model, which can expect the most probable day of the beginning of the crash by analyzing excessive stock price volatility. [ABSTRACT FROM AUTHOR]

Dendranthema grandiflora is an important cut flower with high economic importance in the floriculture industry. Identification of stable and high yielding genotypes of Dendranthema grandiflora, hence becomes paramount for ensuring its year-round production. In this context, the genotype by environment interaction effects on 22 chrysanthemum hybrids across six test environments were investigated. The experiment was conducted using Randomized Complete Block Design with three replications for 6 years and data on various agro-morphological and yield-contributing traits were evaluated. Our analysis revealed significant mean sum of squares due to environmental, genotypic and genotype by environment interaction variations for all examined traits. A 2D GGE biplot constructed using first two principal components computed as 59.2% and 23.3% of the differences in genotype by environment interaction for flower yield per plant. The GGE biplot identified two top-performing genotypes, G2 and G5, while the AMMI model highlighted genotypes G17, G15, G6, G5, and G2 as the best performers. Genotype G17 ranked highest for multiple traits, while G2 displayed high mean flower yield as well as stability across all environments. According to AEC line, genotypes G2 and G5 exhibited exceptional stability, whereas genotypes G4, G18 and G19 demonstrated lower stability but maintained high average flower yields. Hence, our findings provide valuable insights into chrysanthemum hybrids that were not only best performing but also hold promise to meet the growers demand of the cut flower industry and can be recommended for large scale commercial cultivation. [ABSTRACT FROM AUTHOR]

Accurate and reliable mathematical modeling is essential for the optimal control and performance analysis of polymer electrolyte membrane fuel cell (PEMFC) systems, which are mainly implemented based on accurate parameter estimation. In this paper, a multi-strategy tuna swarm optimization (MS-TSO) is proposed to estimate the parameters of PEMFC voltage models and compare them with other optimizers such as differential evolution, the whale optimization approach, the salp swarm algorithm, particle swarm optimization, Harris hawk optimization and the slime mould algorithm. In the optimizing routine, the unidentified factors of the PEMFCs are used as the decision variables, which are optimized to minimize the sum of square errors between the estimated and measured data. The optimizers are examined based on three PEMFC datasets including BCS500W, NedStackPS6 and harizon500W as well as a set of experimental data which are measured using the Greenlight G20 platform with a 25 cm2 single cell at 353 K. It is confirmed that MS-TSO gives better performance in terms of convergence speed and accuracy than the competing algorithms. Furthermore, the results achieved by MS-TSO are compared with other reported approaches in the literature. The advantages of MS-TSO in ascertaining the optimum factors of various PEMFCs have been comprehensively demonstrated. [ABSTRACT FROM AUTHOR]

Frequently orthopedic surgeries require mechanical drilling processes especially for inserted biodegradable screws or removing small bone lesions. However mechanical drilling techniques induce large number of forces as well as have substantially lower material removal rates resulting in prolong healing times. This study focuses on analyzing the impact of quasi-continuous laser drilling on the bone's surface as well as optimizing the drilling conditions to achieve high material removal rates. An ex-vivo study was conducted on the cortical region of desiccated bovine bone. The laser-based drilling on the bovine bine specimens was conducted in an argon atmosphere using a number of laser pulses ranging from 100 to 15,000. The morphology of the resulting laser drilled cavities was characterized using Energy dispersive Spectroscopy (EDS) and the width and depth of the drills were measured using a laser based Profilometer. Data from the profilometer was then used to calculate material removal rates. At last, the material removal rates and laser processing parameters were used to develop a statistical model based on Design of Experiments (DOE) approach to predict the optimal laser drilling parameters. The main outcome of the study based on the laser drilled cavities was that as the number of laser pulses increases, the depth and diameter of the cavities progressively increase. However, the material removal rates revealed a decrease in value at a point between 4000 and 6000 laser pulses. Therefore, based on the sequential sum of square method, a polynomial curve to the 6th power was fit to the experimental data. The predicted equation of the curve had a p-value of 0.0010 indicating statistical significance and predicted the maximum material removal rate to be 32.10 mm3/s with 95%CI [28.3,35.9] which was associated with the optimum number of laser pulses of 4820. Whereas the experimental verification of bone drilling with 4820 laser pulses yielded a material removal rate of 33.37 mm3/s. Therefore, this study found that the carbonized layer formed due to laser processing had a decreased carbon content and helped in increasing the material removal rate. Then using the experimental data, a polymetric equation to the sixth power was developed which predicted the optimized material removal rate to occur at 4820 pulses. [ABSTRACT FROM AUTHOR]

Clustering is an important task in biomedical science, and it is widely believed that different data sets are best clustered using different algorithms. When choosing between clustering algorithms on the same data set, reseachers typically rely on global measures of quality, such as the mean silhouette width, and overlook the fine details of clustering. However, the silhouette width actually computes scores that describe how well each individual element is clustered. Inspired by this observation, we developed a novel clustering method, called SillyPutty. Unlike existing methods, SillyPutty uses the silhouette width for individual elements as a tool to optimize the mean silhouette width. This shift in perspective allows for a more granular evaluation of clustering quality, potentially addressing limitations in current methodologies. To test the SillyPutty algorithm, we first simulated a series of data sets using the Umpire R package and then used real-workd data from The Cancer Genome Atlas. Using these data sets, we compared SillyPutty to several existing algorithms using multiple metrics (Silhouette Width, Adjusted Rand Index, Entropy, Normalized Within-group Sum of Square errors, and Perfect Classification Count). Our findings revealed that SillyPutty is a valid standalone clustering method, comparable in accuracy to the best existing methods. We also found that the combination of hierarchical clustering followed by SillyPutty has the best overall performance in terms of both accuracy and speed. Availability: The SillyPutty R package can be downloaded from the Comprehensive R Archive Network (CRAN). [ABSTRACT FROM AUTHOR]

Objectives: Estimation of body mass from skeletal metrics can reveal important insights into the paleobiology of archeological or fossil remains. The standard approach constructs predictive equations from postcrania, but studies have questioned the reliability of traditional measures. Here, we examine several skeletal features to assess their accuracy in predicting body mass. Materials and Methods: Antemortem mass measurements were compared with common skeletal dimensions from the same animals postmortem, using 115 rhesus macaques (male: n = 43; female: n = 72). Individuals were divided into training (n = 58) and test samples (n = 57) to build and assess Ordinary Least Squares or multivariate regressions by residual sum of squares (RSS) and AIC weights. A leave‐one‐out approach was implemented to formulate the best fit multivariate models, which were compared against a univariate and a previously published catarrhine body‐mass estimation model. Results: Femur circumference represented the best univariate model. The best model overall was composed of four variables (femur, tibia and fibula circumference and humerus length). By RSS and AICw, models built from rhesus macaque data (RSS = 26.91, AIC = −20.66) better predicted body mass than did the catarrhine model (RSS = 65.47, AIC = 20.24). Conclusion: Body mass in rhesus macaques is best predicted by a 4‐variable equation composed of humerus length and hind limb midshaft circumferences. Comparison of models built from the macaque versus the catarrhine data highlight the importance of taxonomic specificity in predicting body mass. This paper provides a valuable dataset of combined somatic and skeletal data in a primate, which can be used to build body mass equations for fragmentary fossil evidence. [ABSTRACT FROM AUTHOR]

High scientific value areas on celestial bodies such as the Moon and Mars are often located in hazardous terrains. To achieve safe landing exploration, a novel planning method is proposed, which can ensure that the planned trajectory maintains a user-specified distance from obstacles, thus reducing potential collision risk induced by factors such as the body size and model uncertainties. Firstly, the basic model of the trajectory optimization problem and its convexification version is given. The obstacles are modeled as polynomial functions, based on which the "if-then" obstacle avoidance logic explicitly considering the safe distance, is described as a sum of squares constraint. This constraint formulation applies to any obstacle described by a finite number of polynomials, independent of the specific expression of the polynomials (reflecting the shape of the obstacle). Subsequently, the convexification process for the obstacle avoidance constraint is given. Finally, the sequential sum of squares programming problem for the obstacle avoidance trajectory is established, which boils down to a series of semidefinite programming problems. Simulation results show that the closest distance between the planned trajectory and obstacles strictly satisfies the specified distance constraint, and the trajectory could avoid non-convex obstacles. With the promising convergence properties of underlying convex optimization algorithms, advanced autonomous obstacle avoidance guidance schemes are expected to be formed based on the proposed trajectory planning method. • Mars/Moon landings cannot ignore unknown and hazardous surface environments. • The trajectory obtained by traditional avoidance methods is too close to obstacles. • The safe distance constraint can be described as a sum of squares constraint. • The proposed planning method applies to any obstacle described by polynomials. [ABSTRACT FROM AUTHOR]

The accurate characterization of typhoon wind profile properties is of great importance in the field of wind engineering and wind design of high-rise structures. In this paper, the average typhoon wind profile characteristics are investigated using the 930 m height measurement data of Typhoon Lekima 2019 obtained from the observations of the mobile acoustic profiling radar deployed in the coastal area. Specifically, this paper adopts a cohesive hierarchical cluster analysis method to classify the mean wind profiles of Super Typhoon Lekima 2019, and the optimal number of clusters is obtained as two classes by the profile coefficient with the sum of squares of clustering errors, the Calinski–Harabasz index, and the Davies–Bouldin index, and the two classes of typical wind profiles are named as cluster 1 type and cluster 2 type. The model fitting analysis of the two types of typical wind profiles was carried out in the height range of 0~300 m after classification, and the effects of fitting the cluster 1-type mean wind profiles with the Vickery model and the Snaiki and Wu model and the cluster 2-type mean wind profiles with the Power-law model, the Log-law model, and the Deaves–Harris and Gryning models were discussed. The results show that the cohesive hierarchical cluster analysis method used in this paper can effectively categorize the mean typhoon profiles. In addition, this paper has some reference significance for future research on the characteristics of measured typhoon wind profiles and engineering applications such as the wind-resistant design of high-rise structures. [ABSTRACT FROM AUTHOR]

In ACM TOMS Algorithm 1012, the DELAUNAYSPARSE software is given for performing Delaunay interpolation in medium to high dimensions. When extrapolating outside the convex hull of the training set, DELAUNAYSPARSE calls the nonnegative least squares solver DWNNLS to compute projections onto the convex hull. However, DWNNLS and many other available sum-of-squares optimization solvers were not intended for usage with many variable problems, which result from the large training sets that are typical in machine learning applications. Thus, a new PROJECT subroutine is given, based on the highly customizable quadratic program solver BQPD. This solution is shown to be as robust as DELAUNAYSPARSE for projection onto both synthetic and real-world datasets, where other available solvers frequently fail. Although it is intended as an update for DELAUNAYSPARSE, due to the difficulty and prevalence of the problem, this solution is likely to be of external interest as well. [ABSTRACT FROM AUTHOR]

To select a stable and best-performing cluster bean line over seasons, an experiment was carried out using five shortlisted advanced breeding lines of vegetable cluster bean in randomized block design with four replications in Kharif and Summer seasons of 2019–20 and 2020–21 at ICAR-Indian Institute of Horticultural Research, Bengaluru. Additive mean effect and multiplicative interaction analysis of variation indicated a significant genotype and environment (G × E) interaction for all the traits. A high environment effect of 33.73% of the total sum of squares was observed for the trait pods per cluster followed by yield per hectare, single pod weight, pods per plant and clusters per plant. The first two interactive principal component axes (IPCA) cumulatively contributed 87.9, 97.3, 94.6, 98.6 and 85.6% variations for yield per hectare, number of clusters per plant, pods per plant, pods per cluster and single pod weight, respectively, leaving a small but significant amount of variation in the third IPCA. A mean versus weighted average of absolute score (WAAS) biplot indicated that genotype IIHRCB 26-2-1 is stable and best for the trait clusters per plant (>20 clusters/plant) while, IIHRCB 22-1-1 is superior and stable (WAAS mean nearly 0.00) for pods per cluster with >5 pods/cluster, pods per plant with >90 pods/plant, single pod weight with >3.0 g and yield per hectare around 24 t/ha. For all the environments, genotype IIHRCB 22-1-1 was found 'all-time winner' for yield per hectare and single pod weight. Based on multitrait stability index, IIHRCB 22-1-1 was found the best performer and the most stable genotype. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

This research intends to build a performance index model of raw water infrastructure mathematically by considering technical, non-technical, and environmental aspects. The research location is in Lombok and the Sumbawa Islands. Data is collected by field surveys and questionnaires that are distributed to 160 respondents related to raw water infrastructure in 21 locations. The methodology consists of Partial Least Squares (PLS) and Generalized Reduced Gradient (GRG). The results show that technical, non-technical, and environmental aspects have a significant influence on the performance index of raw water infrastructure. The structural analysis shows that the technical, non-technical, and environmental variables have a positive and significant influence on the performance index. The performance index of raw water infrastructure is successful enough to be developed and tested by using field data and GRG. The evaluation result shows that the model gives an accurate estimation of raw water infrastructure performance in Nusa Tenggara Barat province. The performance index model for raw water infrastructure is as follows: 0.521 IKTK + 0.305 IKNT + 0.174 IK Li with the sum of square residual (SSR) is 83.21, the root mean square error (RMSE) is 0.44, the mean square error (MSE) is 3.97, and the accuracy level is 95.25%. This research provides the development of an evaluation method for raw water infrastructure performance and a valuable outlook for policymakers in managing and maintaining raw water infrastructure to support sustainable water resources in the future. Considering some aspects of this, it is hoped the efforts to increase the quality of raw water infrastructure can be more directed and effective, contributing to increasing society's prosperity and a sustainable environment in the region. [ABSTRACT FROM AUTHOR]

We study learning from unlabeled data distributed across clients in a federated fashion where raw data do not leave the corresponding devices. We develop a K-means clustering algorithm within this federated setting where the local datasets are clustered at the clients, and a server generates the global clusters after aggregating the local ones. Given the importance of initialization on the federated K ̲ -means algorithm (FKM), our objective is to find better initial centroids by utilizing the local data stored on each client. To this end, we start the centroid initialization at the clients, rather than at the server, since the server initially lacks any preliminary insight into the clients' data. The clients first select their local initial clusters and subsequently share their clustering information (including cluster centroids and sizes) with the server. The server then employs a greedy algorithm to determine the global initial centroids based on the information received from the clients. We refer to this idea as G-FKM. Numerical results obtained from both synthetic and public datasets demonstrate that our proposed algorithm demonstrates accelerated convergence, exhibiting reduced within-cluster sum of squares ( WCSS ¯ ) and higher adjusted Rand Index compared to three distinct federated K-means variants. This improvement comes at a relatively low cost of sending limited additional information from the clients to the server, rather than conducting the initialization entirely at the server. Furthermore, we have also observed that the proposed algorithm performs better than the centralized algorithm for cases where the data distribution across the clients is highly skewed. [ABSTRACT FROM AUTHOR]

This paper describes the development of a dual hazard spectrum for use in the dynamic analysis of steel frames subject to the combined effects of earthquakes and wind. The proposed spectrum is obtained by combining the power spectra of earthquakes and wind using the square root of the sum of squares (SRSS) combination method. An equivalent time excitation function is then computed using an inverse fast Fourier transform (IFFT) and serves as input for the dynamic analysis. Using time-history analysis on the OpenSees platform, the dynamic responses expressed in terms of peak and residual inter-story and roof drifts for two multistory steel frames located in two US cities (Los Angeles and Charleston) are obtained to demonstrate that designing these buildings based on just one hazard may not be adequate. For frames that are considered under-designed, an energy-based design procedure that uses buckling-restrained braces (BRBs) to dissipate the excess energy imparted to these frames is proposed so they will satisfy the FEMA 356 recommended drift limits for the performance levels of immediate occupancy and life safety. [ABSTRACT FROM AUTHOR]

Purpose: The 3D multi-shot EPI imaging offers several benefits including higher SNR and high isotropic resolution compared to 2D single shot EPI. However, it suffers from shot-to-shot inconsistencies arising from physiologically induced phase variations and bulk motion. This work proposed a motion compensated structured low-rank (mcSLR) reconstruction method to address both issues for 3D multi-shot EPI. Methods: Structured low-rank reconstruction has been successfully used in previous work to deal with inter-shot phase variations for 3D multi-shot EPI imaging. It circumvents the estimation of phase variations by reconstructing an individual image for each phase state which are then sum-of-squares combined, exploiting their linear interdependency encoded in structured low-rank constraints. However, structured low-rank constraints become less effective in the presence of inter-shot motion, which corrupts image magnitude consistency and invalidates the linear relationship between shots. Thus, this work jointly models inter-shot phase variations and motion corruptions by incorporating rigid motion compensation for structured low-rank reconstruction, where motion estimates are obtained in a fully data-driven way without relying on external hardware or imaging navigators. Results: Simulation and in vivo experiments at 7T have demonstrated that the mcSLR method can effectively reduce image artifacts and improve the robustness of 3D multi-shot EPI, outperforming existing methods which only address inter-shot phase variations or motion, but not both. Conclusion: The proposed mcSLR reconstruction compensates for rigid motion, and thus improves the validity of structured low-rank constraints, resulting in improved robustness of 3D multi-shot EPI to both inter-shot motion and phase variations. [ABSTRACT FROM AUTHOR]

The equation |$x^2 + 1 = 0\mod p$| has solutions whenever p  = 2 or |$4n + 1$|⁠. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of |$SL_2(\mathbb{R})$|⁠. Since our Poincare series have a nontrivial dependence on their Iwasawa θ -coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work. [ABSTRACT FROM AUTHOR]

This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime.We establish a C2,d generalization of M. Christ's smooth sum of squares theorem, and then use a bootstrap argument with the sum of squares decomposition for matrix functions, obtained in our second paper of this series, to prove a hypoellipticity theorem that generalizes some cases of the results of Christ, Hoshiro, Koike, Kusuoka and Stroock and Morimoto for sums of squares, and of Fedıi and Kohn for degeneracies not necessarily a sum of squares. [ABSTRACT FROM AUTHOR]

We consider proof complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this proof complexity with the normal unary encoding in several refutation systems, based on Resolution and Sherali--Adams. We first consider Res(s), which is an extension of Resolution working on s-DNFs (Disjunctive Normal Form formulas). We prove an exponential lower bound of n \Omega (k)/\mathrm{d}(s) for the size of refutations of the binary version of the k-Clique Principle in Res(s), where s=o((log log n) 1/3) and d(s) is a doubly exponential function. Our result improves that of Lauria et al., who proved a similar lower bound for Res(1), i.e., Resolution. For the k-Clique and other principles we study, we show how lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for the binary version, so we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the (weak) Pigeonhole Principle Bin PHPmn. We prove that for any \delta, \epsilon >0, Bin PHPmn requires refutations of size 2n1 \delta in Res(s) for s =O(log 1 2 \epsilon n). Our lower bound cannot be improved substantially with the same method since for m\geq 2 \surd n\mathrm{l}\mathrm{o}\mathrm{g} n we can prove there are 2O(\surd n\mathrm{l}\mathrm{o}\mathrm{g} n) size refutations of Bin PHPmn in Res(log n). This is a consequence of the same upper bound for the unary weak Pigeonhole Principle of Buss and Pitassi. We contrast unary versus binary encoding in the Sherali--Adams (SA) refutation system where we prove lower bounds for both rank and size. For the unary encoding of the Pigeonhole Principle and the Ordering Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the (weak) Pigeonhole Principle Bin PHPmn requires exponentially sized (in n) SA refutations, whereas the binary encoding of the Ordering Principle admits logarithmic rank, polynomially sized SA refutations. We continue by considering a natural refutation system we call "SA+Squares," which is intermediate between SA and Lasserre (Sum-of-Squares). This has been studied under the name static-LS\infty + by Grigoriev et al. In this system, the unary encoding of the Linear Ordering Principle LOPn requires O(n) rank while the unary encoding of the Pigeonhole Principle becomes constant rank. Since Potechin has shown that the rank of LOPn in Lasserre is O(\surd nlog n), we uncover an almost quadratic separation between SA+Squares and Lasserre in terms of rank. Grigoriev et al. noted that the unary Pigeonhole Principle has rank 2 in SA+Squares and therefore polynomial size. Since we show the same applies to the binary Bin PHPn+1 n, we deduce an exponential separation for size between SA and SA+Squares. [ABSTRACT FROM AUTHOR]

We introduce the binary value principle, which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, based on a well-known hypothesis by Shub and Smale about the hardness of computing factorials, where IPS is the strong algebraic proof system introduced by Grochow and Pitassi [J. ACM, 65 (2018), 37]. Conversely, we show that short IPS refutations of this instance bridge the gap between sufficiently strong algebraic and semialgebraic proof systems. Our results extend to unrestricted IPS the paradigm introduced by Forbes, Shpilka, Tzameret, and Wigderson [Theory Comput., 17 (2021), pp. 1--88], whereby lower bounds against subsystems of IPS were obtained using restricted algebraic circuit lower bounds, and demonstrate that the binary value principle captures the advantage of semialgebraic over algebraic reasoning, for sufficiently strong systems. Specifically, we show the following. (1) Conditional IPS lower bounds: The Shub--Smale hypothesis [Duke Math. J., 81 (1995), pp. 47--54] implies a superpolynomial lower bound on the size of IPS refutations of the binary value principle over the rationals defined as the unsatisfiable linear equation.... Further, the related and more widely known \tau -conjecture [Duke Math. J., 81 (1995), pp. 47--54] implies a superpolynomial lower bound on the size of IPS refutations of a variant of the binary value principle over the ring of rational functions. No prior conditional lower bounds were known for IPS or apparently weaker propositional proof systems such as Frege systems (though our lower bounds do not translate to Frege lower bounds since the hard instances are not Boolean formulas). (2) Algebraic versus semialgebraic proofs: Admitting short refutations of the binary value principle is necessary for any algebraic proof system to fully simulate any known semialgebraic proof system, and for strong enough algebraic proof systems it is also sufficient. In particular, we introduce a very strong proof system that simulates all known semialgebraic proof systems (and most other known concrete propositional proof systems), under the name Cone Proof System (CPS), as a semialgebraic analogue of the IPS: CPS establishes the unsatisfiability of collections of polynomial equalities and inequalities over the reals, by representing sum-of-squares proofs (and extensions) as algebraic circuits. We prove that IPS polynomially simulates CPS iff IPS admits polynomial-size refutations of the binary value principle (for the language of systems of equations that have no 0/1-solutions), over both Z and Q. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a prime. We denote by Irr p (G) the set of irreducible complex characters of G whose degrees are linear or divisible by p, and we write R p (G) to denote the ratio of the sum of squares of irreducible character degrees in Irr p (G) to the sum of irreducible character degrees in Irr p (G). The Itô–Michler Theorem on character degrees states that R p (G) = 1 if and only if G has a normal abelian Sylow p-subgroup. We generalize this theorem as follows: if R p (G) < p + 1 2 , then G has a normal Sylow p-subgroup. [ABSTRACT FROM AUTHOR]

Recently, interest in wireless sensor networks (WSNs) and applications has grown significantly. The path loss caused by buildings and tall trees is a major obstacle in the distribution of sensor nodes. Because of absorption, dispersion, and attenuation by the structures or trees, a degradation occurs in the reliability of the communication link. In this paper, four path-loss models (PLMs) were developed for a LoRa (long-range) wireless sensor node mounted on the drone as a part of a WSN for monitoring air pollution over a distance of approximately 1,000 m utilising the received signal strength indicator (RSSI). Four novel PLMs were created using MATLAB's curve-fitting tools, employing polynomial (POLY), exponential (EXP), Gaussian, and power equations. To improve the accuracy of these equations, their coefficients were fine-tuned using the particle swarm optimisation (PSO) algorithm, resulting in the development of hybrid algorithms—EXP-PSO, POLY-PSO, Gaussian-PSO, and power-PSO. The results demonstrated that PSO-enhanced PLMs achieved an outstanding correlation coefficient (R2) value of 1, indicating their exceptional ability to represent PLMs accurately in the tested air population environment utilising WSNs. Notably, the POLY-PSO algorithm outperformed other algorithms, achieving the lowest root-mean-square error (RMSE) of 1.945 × 10−4 and sum of squares error (SSE) of 1.853 × 10−26. The PLMs based on hybridisation algorithms outperformed existing PLMs regarding the correlation coefficient. These findings hold significant implications for designing and optimising WSNs in air pollution applications, offering valuable insights into radio path-loss modelling in this area. [ABSTRACT FROM AUTHOR]